Part 2

Introduction

While studying rotational motion, we restricted ourselves to simpler situations of rigid bodies. A rigid body generally means a hard solid object having a definite shape and size. But in reality, bodies can be stretched, compressed and bent. Even the appreciably rigid steel bar can be deformed when a sufficiently large external force is applied to it. This means that solid bodies are not perfectly rigid. A solid has a definite shape and size. In order to change (or deform) the shape or size of a body, a force is required

When we pull a helical spring by hanging a mass with it, it will get stretched and hence change its length and shape. As soon as we remove the hanging mass, the spring will instantly regain its original shape and size.

The property of a body, by virtue of which it tends to regain its original size and shape when the applied force is removed, is known as elasticity and the deformation caused is known as elastic deformation.

What are the Mechanical Properties of Solids?

Mechanical properties of solids elaborate the characteristics such as the resistance to deformation and their strength. Strength is the ability of an object to withstand the applied stress, to what extent can it bear the stress. Resistance to deformation is how resistant an object is to the change of shape. If the resistance to deformation is less, the object can easily change its shape and vice versa. Therefore, some of the mechanical properties of solids include:

• Elasticity: When we stretch an object, it changes its shape and when we leave, it regains its shape. Or we can say it is the property to regain the original shape once the external force is removed. Example: Spring

• Plasticity: When an object changes its shape and never comes back to its original shape even when an external force is removed. It is the property of permanent deformation. Example: Plastic materials.
• Ductility: When an object can be pulled in thin sheets, wires, or plates, it has ductile properties. It is the property of being drawn into thin wires/sheets/plates. Example: Gold or Silver
• Strength: The ability to withstand an applied stress without failure. Many categories of objects have higher strength than others.

In this chapter, we will study elasticity in detail

Elastic body and Elasticity

A body that returns to its original shape and size on the removal of the deforming force (when deformed within a certain limit) is called an elastic body.

The property of matter by virtue of which it regains its original shape and size when the deforming force has been removed is called elasticity.   For example: If we stretch a spring and release it, it will regain its original size.

Now let us understand how it happens?

Whenever a load is attached to a thin hanging wire, it elongates and the load moves downward. The amount by which the wire elongates depends on the amount of load and the nature of the material.

The cohesive force between the molecules of hanging mass offers resistance against the deformation and the force of resistance increases with the deformation.

As soon as the deformation force is removed, the wire tries to regain its original shape. Thus we may conclude that if some external force is applied to a body, it has two effects on it.

1.  Deformation of the body.
2. Internal resistance (restoring) forces are developed.

Stress

Whenever an external force is applied to a body, at each cross-section of the body, an internal restoring force is developed which tends to restore the body to its original state. The internal restoring force per unit area of the cross-section of the deformed body is called stress.

It is usually denoted by the σ (sigma)

S.I unit  

Its dimensional formula is

Depending on the ways the deforming force are applied to a body, there are three types of stress

• Longitudinal stress / Normal stress
• Shearing stress
• Volume stress

Normal stress

It is defined as the restoring force per unit area acting perpendicular to the surface of the body. It is of two types

Longitudinal or Tensile stress

When two equal and opposite forces are applied at the end of a circular rod as shown in the figure and increase its length, a restoring force equal to the applied force F normal to the cross-section of the rod comes into existence. This restoring force per unit area of cross-section is known as tensile stress.

Tensile stress = F/A 

Consider a rod of length l, Force F area applied on it due to which the final length of the rod becomes L+ΔL. thus, the increase in length ΔL.

Compressive stress

When two equal and opposite forces are applied at the ends of a rod as shown in the figure to decrease in length or compress it, then again restoring force equal to the applied force F comes into existence. This restoring force per unit area of the cross-section of the rod is known as compressive stress.

Compressive stress = F/A

Consider the length of the rod is L, when two equal force is applied, the final length of the rod becomes L-ΔL. thus, and decrease in length of the rod isΔL.

Under tensile stress or compressive stress, the net force acting on an object is zero but the object is deformed. Tensile stress and compressive stress are also terms as longitudinal stress.

Shearing or Tangential stress

When a deforming force acts tangentially to the surface of a body, it produces a change in the shape of the body without any change in volume. This tangential force applied per unit area is equal to the tangential stress.

Tangential stress = F/A

In the case of tangential stress, the deforming force F is applied on a top surface of the cubical body in a tangential direction due to which the upper face is deformed by an angle θ, from its original position is shown in fig.

Volumetric Stress or Bulk Stress

If a body is subjected to a uniform force from all sides, then the corresponding stress is called volumetric stress. There is a change in the volume of the body but not a change in a geometric shape.

Bulk / volumetric stress = F/A

In the case of hydraulic stress, the force applied is perpendicular to every point on the surface due to which a change in volume ΔV  of a body occurred.

Strain

When a deforming force acts on a body, the body undergoes a change in its shape and size. The ratio of the change in configuration of the body to the original configuration is called strain.

Strain =

If there is a change in any of the configuration of the body due to applied deforming force on it, then the body is said to be stained or deformed.

Since strain is the ratio of two like quantities, it is dimensionless and has no units.

According to a change in configuration, change in length, volume, or shape of the body, the strain can be classified as

• Longitudinal strain
• Volumetric strain
• Shear strain

1. Longitudinal strain: It is defined as the change in length per unit of original length when the body is deformed by external forces.

2.  Volumetric strain:   It is defined as the change in volume per unit of original volume when the body is deformed by external forces.    Volumetric strain = change in volume original volume =ΔV/V
3. Shear strain:   It is the deforming forces that produce a change in the shape of the body, then the strain is called shear strain. It is defined as the angle in radians through which a plane perpendicular to the fixed surface of the cubical body gets turned under the effect of tangential force.

The angle θ is called the angle of shear.

Shear  strain  ,θ = tan θ=Δ X/L

Hooke's Law is a principle of physics that states that the force needed to extend or compress a spring by some distance is proportional to that distance. The law is named after 17th-century British physicist Robert Hooke, who sought to demonstrate the relationship between the forces applied to a spring and its elasticity.

Hooke’s law states that the extension produced in the wire is directly proportional to the load applied within the defined limit of elasticity.

Extension produced α load applied 

Later on, it was found that this law is applicable to all types of deformation such as compression, bending, twisting etc.  and thus a modified form of Hooke’s law was given as,

Within the elastic limit, the stress developed is directly proportional to the strain produced in a body.

Stress α   strain    ⇒  Stress = E ×Strain

Or,   E= stress / strain = constant ( modulus of elasticity )

For example: when we apply force on a spring, it gets compressed/ elongated and a restoring force is produced opposite to its displacement.

Restoring force  F= -k x  ( is proportional to displacement)

• Hooke’s law is valid only in the linear portion of the stress-strain curve. This law is not valid for large values of strains.
• Modulus of elasticity ‘E’ depends on the nature of the material of the body and is independent of its dimensions like length, area, volume etc.

Stress-strain Relationship

When a wire is stretched by an applied force, then a typical graph is obtained (especially in the case of metals) as shown below.

• OA is a straight line showing that the material follows Hooke’s law. Point A is called the proportionality limit (σp).  Beyond this, stress and strain do not exhibit a linear relationship.
• Up to B, if the load is removed, the material will regain its original shape. So material shows elastic behavior up to point B.

Thus curve OB represents the elastic curve. Point B is called yield point and corresponding stress is called yield strength (σy).

• Beyond point B, the strain increases rapidly even for a small change in stress or load. And when the load is removed between points B and D, the body does not regain its original dimensions. So, even when the stress (or load) is zero, there remains some strain on the material. The deformation is called plastic deformation.
• Point D on the graph represents the ultimate tensile strength (σu) of the material. Beyond this, additional strain is produced even by reduced load, where the fracture occurs is known as Fracture point E.

Modulus of Elasticity

The modulus of elasticity or coefficient of elasticity of a body is defined as the ratio of stress to the corresponding strain within the elastic limit.

Modulus of Elasticity = Stress/ strain

S.I. unit of Modulus of Elasticity is N/m² or Pascal (Pa) and its dimension is [ML-¹T-²].

There are three types of modulus of elasticity:

1.  Young’s Modulus  ( Y)
2. Bulk modulus  ( B)
3. Modulus of rigidity or shear modulus (G)

1. Young’s Modulus

Within the elastic limit, the ratio of longitudinal stress to the longitudinal strain is called Young’s Modulus of the material of the wire.

Young's modulus (Y) is a property of the material that tells us how easy it can stretch and deform and is defined as the ratio of tensile stress (σ) to tensile strain (ε). Where stress is the amount of force applied per unit area (σ = F/A) and strain is extension per unit length (ε = dl/l).  young's modulus, Y= longitudinal stress/ longitudinal strain.   

longitudinal stress= F/A   ; Longitudinal strain = ΔL/L ,

,

Where F= load, A= area of cross-section, L= Length of rod

For metals Young’s moduli are large. Therefore, these materials require a large force to produce a small change in length. 

Steel has a larger value of young’s modulus than copper, brass and aluminum. Steel is more elastic than copper, brass and aluminum. It is for this reason that steel is preferred in heavy-duty machines and in structural designs. Wood, bone, concrete and glass have rather small Young’s moduli.

If the extension is produced by a load of mass ‘m’ then,    F= mg.

Here the wire has a circular cross-section. So area of cross-section  A=πr² 

So the formula for young’s modulus can be written as

2. Bulk Modulus of Elasticity

Within the elastic limit, the ratio of normal stress to the volumetric strain is called the bulk modulus of elasticity.

Bulk modulus, B  = Normal stress / volumetric strain

So,   B= -pV/ΔV

What is meant by bulk modulus?

Sometimes referred to as incompressibility, the bulk modulus is a measure of the ability of a substance to withstand changes in volume when under compression on all sides. It is equal to the quotient of the applied pressure divided by the relative deformation.

The units for the bulk modulus are Pascal’s (Pa) or newtons per square meter (N/m²) in the metric system.

Compressibility: The reciprocal of the bulk modulus of a material is called its compressibility.

compressibility  K= 1/B = -ΔV/pV

Bulk modulus is used to measure how incompressible a solid is. Besides, the more the value of B for a material, the higher is its nature to be incompressible. For example, the value of B for steel is  1.6×10¹¹  N/m² and the value of B for glass is 4×10¹⁰ N/m²   Here, K for steel is more than three times the value of K for glass. This implies that glass is more compressible than steel.

3. Modulus of rigidity or shear modulus

Within the elastic limit, the ratio of tangential stress (shear stress) to shear strain is called the modulus of rigidity of shear modulus.  It is denoted by G or η.

To measure the stiffness of materials, the shear modulus is one of many quantities. The deformation of a solid is concerned with the shear modulus when it experiences a force parallel to one of its surfaces while its opposite face experiences an opposing force.

Let us consider a cube whose lower face is fixed and a tangent force F acts on the upper face whose area is A, as shown in the figure.

Tangential stress= F/A

Let the vertical sides of the cube shit through an angle θ, called shear strain

Therefore, the Modulus of rigidity is given by 

According to the diagram,   tan θ= ΔX/L   also  θ≈tan θ  for small θ

Poisson’s Ratio

When a material is stretched in one direction, it tends to compress in the direction perpendicular to that of force application and vice versa. The measure of this phenomenon is given in terms of Poisson’s ratio. For example, a rubber band tends to become thinner when stretched.

What is the position ratio?

Poisson's ratio is “the ratio of transverse contraction strain to longitudinal extension strain in the direction of the stretching force.” Here, Compressive deformation is considered negative. Tensile deformation is considered positive.

Therefore Poisson’s ratio σ is given by,

Poisson's ratio is the ratio of lateral strain to longitudinal strain. It has no units.

Relation between various modulus of Elasticity

The elastic moduli of a material, like Young’s Modulus, Bulk Modulus, and Shear Modulus are specific forms of Hooke’s law, which states that for an 

elastic material, the strain experienced by the corresponding stress applied is proportional to that stress. Thus, we can write the relation between elastic constants by the following equation:

2 G (1+σ)= Y= 3B( 1-2σ)

Where,

• G is the Shear Modulus
• Y is the Young’s Modulus
• B is the Bulk Modulus
• σ  is Poisson’s Ratio

We can derive the elastic constant’s relation by combining the mathematical expressions relating to terms individually.

• Young modulus can be expressed using Bulk modulus and Poisson’s ratio as –

Y= 3B( 1-2σ)

• Similarly, Young’s modulus can also be expressed using rigidity modulus and Poisson’s ratio as-

Y= 2G (1+ σ)

• Combining the above two-equation and solving them to eliminate Poisson’s ratio we can get a relation between Young’s modulus and bulk modulus B  and modulus of rigidity as -

Y=  9BG/(G+3B)

A fun thing to try: Virtual lab

Below is the link to the simulation of Hooke’s law

Hooke's law

In this simulation, we have three parts:  Intro, systems, and energy

1. Intro: In the Intro, we have a spring whose one end is fixed and the other end can be pulled.

• We can fix the spring constant of the spring and then by changing the applied force we can see how much displacement it is producing in the spring.
• We can then change the spring constant of the spring and do the same process again.
• We will come to know that it is easier to stretch a spring with lower spring constant.

2. System: In the system we have the option of parallel combination of spring and a series combination of spring. We can choose any of them.

• We can then fix the spring constant of the springs and apply force and see how the displacement is changing with the applied force.
• Now we can change the spring constant and do the same process.
• This should be done with both combinations of spring: Series and parallel.

3. Energy: In this part we can see the value of potential energy of the spring.

• We can fix the spring constant of the spring and then change the displacement and see how the potential energy is varying.
• We can also fix the displacement and vary spring constant and see how the potential energy of the spring is changing.

Introduction

Thermodynamics is that branch of physics which deals with concepts of heat and temperature and their relation to energy and work.

We can also consider it as a macroscopic science which deals with bulk systems and tells us about the system as a whole.

The foundation of thermodynamics is the conservation of energy and the fact that the heat flows spontaneously from hot to the cold body and not the other way around. The study of heat and its transformation to mechanical energy is called thermodynamics. It comes from a Greek word meaning “Movement of heat”.

A collection of large numbers of molecules of matter (solid, liquid or gas) that are arranged in a manner such that these possess particular values of pressure, volume and temperature form a thermodynamic system.

The distinction between mechanics and thermodynamics is worth bearing in mind. In mechanics, our interest is in the motion of particles or bodies under the action of forces and torques. Thermodynamics is not concerned with the motion of the system as a whole. It is concerned with the internal macroscopic state of the body.

In this chapter, we will learn about the laws of thermodynamics which describes the system in terms of macroscopic variables, and reversible and irreversible processes. Finally, we will also learn on what principle heat engines, refrigerators and Carnot engines work.

Thermal equilibrium

Equilibrium in mechanics means that the net external force and torque on a system are zero. The term ‘equilibrium’ in thermodynamics appears in a different context: we say the state of a system is an equilibrium state if the macroscopic variables that characterize the system do not change in time.

For example, a gas inside a closed rigid container, completely insulated from its surroundings, with fixed values of pressure, volume, temperature, mass and composition that do not change with time, is in a state of thermodynamic equilibrium

Consider two bodies at different temperatures one is at 30 C and another at 60 C then the heat will flow from the body at a higher temperature to the body at a lower temperature. Heat will flow till both bodies acquire the same temperature. This state when there is no heat flow between two bodies when they acquire the same temperature is known as thermal equilibrium.

Types of Equilibrium

Thermal Equilibrium: - Two systems are said to be in thermal equilibrium with each other if the temperatures of both systems do not change with time.

Chemical Equilibrium: - Two systems are said to be in chemical equilibrium with each other if the composition of the system does not change over time.

Mechanical Equilibrium: - Two systems are said to be in mechanical equilibrium with each other if the pressure of the system doesn’t change with time.

A system is said to be in Thermodynamic equilibrium when all of its macroscopic variables are constant.

System and surrounding

System: - System is defined as any part of the universe enclosed by some boundary through which exchange of heat or energy takes place.

Surroundings: - Any part of the universe which is not a system. Systems and surroundings constitute the Universe.

For example: -

1. If we consider hot coffee in a kettle then the kettle is the system and everything else is the surroundings.

2. A cup of hot coffee after some time becomes cold due to the exchange of heat between the system and surroundings.

Types of system

A thermodynamic system is a specific portion of matter with a definite boundary on which our attention is focused. There are three types of systems:

• Isolated System – An isolated system cannot exchange energy and mass with its surroundings. The universe is considered an isolated system. For example a thermos flask
• Closed System – Across the boundary of the closed system, the transfer of energy takes place but the transfer of mass doesn’t take place.

For example A balloon filled with gas, A pot with a lid etc.

• Open System – In an open system, the mass and energy both may be transferred between the system and surroundings.

For Example: - Water boils in a pan without lid, a cup of coffee etc.

Types of walls

Adiabatic wall: - It is an insulating wall that doesn’t allow heat to flow from one system to another. This means the temperature of both the systems won’t change with time.

Consider 2 systems A and B as shown in the figure, which are separated by adiabatic walls. Let the pressure and volume of A be (P1, V1) and (P2, V2).

Both these systems are also separated from the surroundings by an adiabatic wall which means there is no flow of heat between A and surroundings and also B and surroundings.

For example: - Thermos Flask. In which tea or coffee remains hot for a long time as it is made of insulating walls due to which there is no heat flow between tea and surroundings.

Diathermic (conducting) Wall: - It is a conducting wall that allows the flow of heat between any 2 systems.

• Consider two systems A and B which are separated by a conducting wall. System A is at higher temperature T1, pressure P1 and volume V1 and System B is at lower temperature T2, pressure P2 and volume V2.

• There is flow of heat from a system at a higher temperature to the system at a lower temperature till the systems reach thermal equilibrium.

For Example: - A vessel made up of metals like copper or aluminum has diathermic / conducting walls

Zeroth law of thermodynamics: Temperature

Zeroth law of thermodynamics states that when two systems are in thermal equilibrium through a third system separately then they are in thermal equilibrium with each other also.

Forge: - Consider two systems A and B which are separated by an adiabatic wall. Heat flow happens between systems A and C, and between B and C, due to which all 3 systems attain thermal equilibrium.

Systems A and B are in thermal equilibrium with C. Then they will be in equilibrium with each other also.

• Zeroth's Law of Thermodynamics suggested that there should be some physical quantity that should have the same value for the system to be in thermal equilibrium.
• This physical quantity that determines whether a system is in equilibrium or not is Temperature.
• Temperature is the quantity that determines whether the system is in thermal equilibrium with the neighboring system.
• When the temperature becomes equal then the flow of heat stops.

Thermodynamic state variable

Thermodynamic state variables are the macroscopic quantities which determine the thermodynamic equilibrium state of a system. These macroscopic quantities are known as thermodynamics state variables.

• As they determine the state of the system, that is pressure, volume and temperature, at one particular time they are known as thermodynamic state variables. Pressures (P), Volume (V), Temperature (T), mass (m), and Internal energy (U) are the thermodynamic state variables.
• These variables can tell us the position or the condition of any gas at that particular time.
• A system not in equilibrium cannot be described by state variables. It means the macroscopic variables are changing with time and they are not constant.

Types of thermodynamic state variables:-

1. Extensive variables: - They indicate the size of the system, which means extensive variables are those that depend on the mass of the system or the number of particles in the system.  Example: volume, mass, internal energy. If we consider a system whose mass is greater than the size of that system is greater. All these depend on the size of the system.
2. Intensive variables: -A quantity in a macroscopic system that has a well-defined value at every point inside the system and that remains (nearly) constant when the size of the system is increased. Examples of intensive variables are pressure, temperature, density, specific heat capacity at constant volume, and viscosity.

In the figure given below various examples of extensive and intensive variables are given.

Internal Energy

It is defined as the sum of kinetic energies and potential energies of the molecules constituting the system as a whole and not of individual molecules. It is a macroscopic variable of the system.

It is denoted by U.  It is an extensive thermodynamic state variable as it depends on the size of the system.

It only depends on the state of the system at that particular time and does not depend on how the system has reached that state.

There are two modes of changing the internal energy of a system

1. Heat
2. Work

Consider again, for simplicity, the system to be a certain mass of gas contained in a cylinder with a movable piston. There are two ways to change the internal energy of the system.

By Heat: One way is to put the cylinder in contact with a body at a higher temperature than that of the gas. The temperature difference will cause a flow of energy (heat) from the hotter body to the gas, thus increasing the internal energy of the gas.

By Work: The other way is to push the piston down i.e. to do work on the system, which again results in increasing the internal energy of the gas.

Both these things could happen in the reverse direction. With surroundings at a lower temperature, heat would flow from the gas to the surroundings. Likewise, the gas could push the piston up and do work on the surroundings.
Few things to be remembered:

The notion of heat should be carefully distinguished from the notion of internal energy. Heat is certainly energy, but it is the energy in transit. The state of a thermodynamic system is characterized by its internal energy, not heat.

A statement like ‘a gas in a given state has a certain amount of heat’ is as meaningless as the statement that ‘a gas in a given state has a certain amount of work’. In contrast, a gas in a given state has a certain amount of internal energy is a perfectly meaningful statement. Similarly, the statements ‘a certain amount of heat is supplied to the system’ or ‘a certain amount of work was done by the system’ are perfectly meaningful.

First Law of thermodynamics

According to the first law of thermodynamics: - The change in the internal energy of a closed system is equal to the amount of heat supplied to the system, minus the amount of work done by the system on its surroundings.
Mathematically, ΔQ=ΔU+ΔW=ΔU+PΔV

The First law of thermodynamics is the same as the law of conservation of energy. According to the law of energy conservation: - Energy can neither be created nor be destroyed, only transformed to other forms.

Sign conventions:

• When the heat gets supplied to the system, then ΔQ is taken positive and when heat gets withdrawn from the system, ΔQ is taken negative. 
• When a gas expands, work done by the gas is taken positive whereas when a gas contracts, work is taken negatively. Work done is also a path variable so its value also depends on the path chosen.
•  ΔU is taken positively when temperature increases while ΔU is taken negative when temperature decreases.

Remember, Heat  ΔU  and work done is a path variable so it depends on the path chosen. Internal energy is a state variable so its value doesn’t depend on the path followed but only depends on the initial and final state of the system.

Limitations of the first law of thermodynamics

The first law of thermodynamics plays an important role in thermodynamics as it can be applied to know how much work will be obtained by transferring a certain amount of heat energy in a given thermodynamics process. However, the first law of thermodynamics suffers from the following limitations.

• First law of thermodynamics does not indicate the direction of heat transfer.
• First law of thermodynamics does not tell anything about the conditions under which heat can be transformed into work.
• The first law does not indicate why the whole of the heat energy cannot be continuously converted into mechanical work.

Introduction

Liquids and gases can flow and are therefore called fluids. It is this property that distinguishes liquids and gases from solids in a basic way. Fluids are everywhere around us. Earth has an envelope of air and two-thirds of its surface is covered with water. Water is not only necessary for our existence; every mammalian body consists mostly of water. All the processes occurring in living beings including plants are mediated by fluids. Thus understanding the behaviour and properties of fluids is important.

How are fluids different from solids? What is common in liquids and gases?

Solids have fixed volume and shape, liquids have fixed volume but not fixed shape. Gases on the other hand neither have fixed volume nor fixed shape.

Fluids can be defined as any substance which is capable of flowing.

They don’t have any shape of their own. For example:-water which does not have its own shape but it takes the shape of the container in which it is poured. But when we pour water in a tumbler it takes the shape of the tumbler

Fluids are assumed to be the incompressible (i.e., the density of liquid is not dependent on the variation in pressure and remains constant).

Fluids are also assumed to be non-viscous (i.e., the two liquid surfaces in contact are not pressing any tangential force on each other)

Pressure

Pressure is defined as the physical force exerted on an object. The force applied is perpendicular to surface of objects per unit area.

Mathematically ,  P= F/ A

Unit of pressure is Pascals (Pa).

Since P= F/A   so pressure is inversely proportional to the Area.  If the area would be less, pressure would be more.

For Example:-

Consider a very sharp needle which has a small surface area and consider a pencil whose back is very blunt and has more surface area than the needle.

If we poke a needle in our palm it will hurt as the needle gets pierced inside our skin. Whereas if we poke the blunt side of the pencil into our hand it won’t hurt so much. This is because the area of contact between the palm and the needle is very small therefore the pressure is large.

Whereas the area of contact between the pencil and the palm is more therefore the pressure is less.

Conclusion: Two factors which determine the magnitude of the pressure are:-

• Force – greater the force greater is the pressure and vice-versa.
• Coverage area –greater the area less is the pressure and vice-versa.

Fluid pressure

Normal force exerted by fluid per unit area. This means force is acting perpendicular to the surface of contact.

If a body is submerged in the water, force is exerted by the water perpendicular to the surface of the body. Fluid force exerts itself perpendicularly to any surface in the fluid, no matter the orientation of that surface. Thus, fluid pressure has no intrinsic direction of its own and can be considered as a scalar quantity.

Pressure is a scalar quantity. Because the force here is not a vector quantity but it is the component of force normal to the area.

Dimensional formula for pressure is[ ML^-1T²]. The S.I unit is Pascal (Pa).

Atmospheric pressure: The atmospheric pressure at a point is equal to the weight of the column of air of unit cross-sectional area extending from that point to the top of the atmosphere. Its value is 1.013 ×10⁵ Pa at sea level. Atmospheric pressure drops as altitude increases. It is measured using an instrument called a barometer.

Definition of 1 atm

An atmosphere (atm) is a unit of measurement equal to the average air pressure at sea level at a temperature of 15 degrees Celsius (59 degrees Fahrenheit). One atmosphere is 1,013 millibars, or 760 millimetres (29.92 inches) of mercury. Atmospheric pressure drops as altitude increases

Pascal’s law

Pascal’s law states that if the pressure is applied to uniform fluids that are confined, the fluids will then transmit the same pressure in all directions at the same rate.

Pascal’s law holds good only for uniform fluids.

Let us try to understand this with a suitable example. Consider a vessel of circular shape filled with water which has 4 openings and in the entire openings 4 pistons are attached.

• Apply force on the first piston; this piston will move inward and all other pistons will move outwards.

• This happens because when this piston moves inwards the pressure is exerted on the water. Water transmits this pressure in all the directions.
• The other pistons, except A, moves at the same speed which shows water has exerted pressure in all the directions

Conclusion:-

1. For a uniform fluid in equilibrium, pressure is the same at all points in a horizontal plane. This means there is no net force acting on the fluid; the pressure is the same at all the points.

1. A fluid moves due to the differences in pressure. That means fluid will always move from a point which is at a higher pressure to the point which is at a lower pressure.

Archimedes principle

Archimedes Principle:

• Consider a body partially or fully dipped in a fluid. The fluid exerts a contact force on this body. The resultant of all these contact forces is termed buoyant force or up thrust.
• F=weight of fluid displaced by the body
• This force is termed buoyant force and it acts vertically upwards (opposite to the weight of the body) through the centre of gravity of the displaced fluid. Mathematically,       

F=Vσg

Where V is the volume of displaced liquid and σ is the density of the liquid

• The apparent reduction in weight of body =Up thrust = weight of liquid displaced by the body.

Variation of pressure with depth

Consider a cylindrical object inside a fluid, consider two positions for this object. Fluid is at rest therefore the force along the horizontal direction is zero.

Force along the vertical direction

Consider two positions 1 and 2. Force at position 1 is perpendicular to cross-sectional area   A₁F₁ = P₁

Similarly,  A₂ F₂=P₂ 

Totalthe  force   F_net= F₁+F₂   as F₁ is along negative y axis and is negative and F₂ is along positive y axis and hence positive so we have now,

 F_net= -F₁+F₂ = -P₁A₁+P₂A₂=(P₂-P₁)A    , Taking A₁=A₂=A

This net force will be balanced by the weight of the cylinder. Therefore under equilibrium conditions.

Fnet= mg (weight of cylinder)=ρ Vg( weight of liquid displaced)

Fnet= (P₂-P₁) A=ρg(Ah)   using V= Ah

Which gives  (P₂-P_{1)A =ρghA   ⇒ (P2-P1)=ρgh}

Therefore the difference in the pressure is dependent on the height of the cylinder.

Consider the top of the cylinder exposed to air therefore P₁ = P_a (atmospheric pressure)

Then   P₂= P_a+ρgh

Conclusion:

•  The pressure P₂ , at depth  below the surface of a liquid open to the atmosphere is greater than atmospheric pressure by an amount ρgh. 
• The pressure is independent of the cross sectional or base area or the shape of the container.
• Thus, the pressure P, at depth below the surface of a liquid open to the atmosphere, is greater than atmospheric pressure by an amount ρgh. The excess of pressure, P − Pa, at depth h is called a gauge pressure at that point.

Hydrostatic Paradox

Hydrostatic Paradox means: - hydro = water, static =at rest

Paradox means that something is taking place surprisingly.

• Consider 3 vessels of very different shapes (like thin rectangular shape, triangular and some filter shape) and we have a source from which water enters into these 3 vessels.
• Water enters through the horizontal base which is the base of these 3 vessels. We observe that the level of water in all the 3 vessels is the same irrespective of their different shapes.
• This is because pressure at some point at the base of these 3 vessels is the same.
• The water will rise in all these 3 vessels till the pressure at the top is same as the pressure at the bottom.
• As pressure is dependent only on height therefore in all the 3 vessels the height reached by the water is the same irrespective of difference in their shapes.

This experiment is known as Hydrostatic Paradox.

Applications: Pascal’s law for transmission of fluid pressure

Hydraulic lift:-

Hydraulic lift is a lift which makes use of fluid.  For example: Hydraulic lifts that are used in car service stations to lift the cars.

Principle: -

• Inside a hydraulic lift there are 2 platforms, one has a smaller area and the other one has a larger area. It is a tube-like structure which is filled with uniform fluid.
• There are 2 pistons (P1 and P2) which are attached at both the ends of the tube. Cross-sectional area of piston P1 is A1 and piston P2 is A2.
• If we apply force F1 on P1, pressure gets exerted and according to Pascal’s law pressure gets transmitted in all the directions and same pressure gets exerted on the other end. As a result the Piston P2 moves upwards.

Advantage of using hydraulic lift is that by applying small force on the small area we are able to generate a larger force.

Mathematically ,  F₁= P₁A₁  and F₂= P₂A₂    

Since by Pascal’s law

Hydraulic Brakes

• Hydraulic brakes work on the principle of Pascal’s law.
• According to this law whenever pressure is applied on fluid it travels uniformly in all directions.
• Therefore when we apply force on a small piston, the pressure gets created which is transmitted through the fluid to a larger piston. As a result of this larger force, uniform braking is applied on all four wheels.
• As braking force is generated due to hydraulic pressure, they are known as hydraulic brakes.
• Liquids are used instead of gas as liquids are incompressible.

Effect of gravity on fluid pressure

We can start by stating the relationship between gravity and fluid pressure. We can define both of the terms. We can also write down the formula to find the fluid pressure and see if it is related to gravity. Gravity is a force existing between bodies.

The formula to find the fluid pressure is given by the formula,

P=ρgh

Where ρ is the density of the fluid, g is the acceleration due to gravity and h is the depth of the fluid level.

Fluid pressure is the pressure at a point within a fluid arising due to the weight of the fluid. Gravity is the universal force of attraction acting between all matters.

Therefore, according to the formula   P=ρgh  the pressure exerted by a fluid, is directly proportional to the specific gravity at any point and to the height of the fluid above the point.

A fun thing to do: Virtual lab

Below is the link of the simulation of under pressure

Under pressure, this simulation is to understand the concept of pressure.  It will help us understand how pressure varies with depth for different fluids.

What can we do in this simulation?

• We can choose the density of the liquid and can fill the liquid up to the height we want to fill it.
• We can also change the value of gravity and using the meter to measure pressure we can get the value of pressure in terms of Metric unit (Pa) and other units like ‘atm’. We just need to drag the meter and place it to the point where we want to measure the pressure.
• We can also on /off the atmospheric pressure in this simulation to see its effects.

Introduction

In this chapter, we will study some of the thermal properties of matter.

This topic discusses various thermal phenomena and how a matter behaves when subjected to the flow of thermal energy. We are specifically concerned in

• Thermal expansion
• Heat and calorimetry
• Transfer of heat

We all have common-sense notions of heat and temperature. Temperature is a measure of ‘hotness’ of a body. A kettle with boiling water is hotter than a box containing ice. In physics, we need to define the notion of heat, temperature, etc., more carefully.

When the body is heated, various changes take place. It could expand, it can become hotter, it can change phase etc.  Temperature is a measure of the hotness of a body. When water boils or freezes, its temperature does not change during these processes even though a great amount of heat is flowing into or out of it.

You might have noticed that you feel hotter on a sunny afternoon as compared to a windy night. This is because of the difference in temperatures. Temperature is very high in the afternoon as compared to night. This chapter basically gives us the information about thermal properties of matter where we will study about the properties of different substances by virtue of heat/heat transfer.

Temperature and heat

Temperature is the relative measure or indication of the hotness and coldness of a body. A hot cooker is said to have higher temperatures and ice cubes to have lower temperature. An object at a higher temperature is said to be hotter than the one at a lower temperature. The S.I unit of Temperature is Kelvin (K).

A cup of hot soup and cold ice cream.

Heat

When we put a cold spoon into a cup of hot tea, the spoon warms up and the tea cools down as they were trying to equalize the temperature. Energy transfer that takes place solely because of temperature difference is called heat flow of heat transfer and the energy transferred is called heat.  The S.I. unit of heat transfer is expressed in Joule (J).

Measurement of Temperature

A physical property that changes with temperature is called thermometric property. When a thermometer is put in contact with a hot body, the mercury expands, increasing the length of the mercury column, which can be calibrated and later be used to measure temperature.

This was one such example, there are many such which enable us to measure temperature.

There are three scales of measurement of Temperature.

• Celsius scale
• Fahrenheit scale
• Kelvin scale

The standard scale of measurement of temperature is Kelvin scale.

1. Celsius scale:   It defines the ice point at 0 degree Celsius and the steam point temperature as 100 degree Celsius. The space between 0 and 100 degree Celsius is equally divided into 100 intervals.
2. Fahrenheit Scale: It defines the ice-point temperature as 32 F and the steam point is 212 F. The space between 32 F and 212 F is divided into 180 intervals.
3.  Kelvin scale:  Kelvin scale is a scale of measuring temperature, the melting point of ice is taken as 273 K and the boiling point of water at 373 K. The space between these is divided in 100 intervals. This is also known as the absolute scale of temperature as it has only positive values of Temperature. This scale has been adopted as the standard scale of measuring Temperature.

To convert a temperature from one scale to the other, we must take into account the fact that the zero temperatures of the two scales are not the same. Below is the relation between different scales of temperature.

Ideal gas equations and absolute Temperature

We have Liquid-in-glass thermometers like mercury thermometers, these thermometers do not give accurate readings for temperature other than the ice point and boiling point because of differing expansion properties of liquid.

A thermometer that uses a gas however gives the same readings regardless of which gas is used. This is considered to be a more accurate thermometer than liquid-in-glass thermometer.  Experiments show that all gases at low densities exhibit the same expansion behavior. The variables that describe the behavior of a gas of given quantity (mass) are

1. Pressure
2. Temperature
3. Volume

There are some laws that are followed by gases of low density. These laws are:-

1.  Charles’s law:  This law states that at constant pressure, volume and temperature of the gas are directly proportional for a fixed quantity of gases.

V α T  at constant P    or  V/T= constant

1.  Boyle’s law: This law states that at a constant temperature, the volume of the gas is inversely proportional to the pressure of gas for a fixed quantity of gases.

V α 1/P    at constant Temperature T  ;   PV= constant

2.  Avogadro law: At constant Pressure and Temperature, equal volume of gases contains equal number of molecules of gas. In other words, we can say that at constant P and T, the Volume of the gas is directly proportional to the number of molecules of gas.

At constant  P and T,    V α n  

If we combine these three laws we will get   V α   n T/P; PV  α  nT

To remove the proportionality sign we add a constant R.  PV= nRT

Above equation is called the ideal gas equation and R= constant of proportionality is called the gas constant.   R= 8.31 J mol-¹ K-¹

Absolute zero temperature

This point, where all the atoms have been completely stopped relative to each other, is known as "absolute zero" and corresponds to the number zero on the Kelvin temperature scale. An object cannot be cooled below this point because there is no atomic thermal motion left to stop.

Can absolute zero ever be reached?

Physicists acknowledge they can never reach the coldest conceivable temperature, known as absolute zero.

The zero on a Kelvin scale is called the absolute zero.  Absolute Temperature is equal to minus 273-degree Celsius or 459.67 degrees Fahrenheit.

Thermal expansion

Thermal expansion is the tendency of matter to change in shape, volume, and area in response to a change in temperature. Temperature is a monotonic function of the average molecular kinetic energy of a substance.

Thermal expansion is caused by heating solids, liquids or gases, which makes the particles move faster or vibrate more (for solids). This means that the particles take up more space and so the substance expands

The amount by which it expands depends on three factors: its original length, the temperature change, and the thermal (heat) properties of the metal itself. Some substances simply expand more easily than others.

Thermal expansion is of three types:

• Linear expansion. The expansion in length is called linear expansion.

• Area expansion. The expansion in area is called area expansion.

• Volume expansion. The expansion in volume is called volume expansion.      

If the coefficient of linear expansion is denoted by α

Coefficient of area expansion is denoted by β

Coefficient of volume expansion is denoted by γ

The relation between    α, β and γ is stated as   β= 2 α      and     γ= 3α

so , α : β: γ = 1 :2 :3

Anomalous behavior of water

Water shows some exceptional behavior that is when it is heated at 0°C, it contracts instead of expanding and it happens till it reaches 4 °C. The volume of a given amount of water is minimum at 4 °C therefore its density is maximum (Refer the Fig). After 4 °C water starts expanding. Below 4 °C, the volume increases, and therefore the density decreases. This means water has a maximum density at 4 °C.

Density of water is maximum and the volume of water is minimum at 4 degree Celsius. This is anomalous behavior of water. Because of this property of water in lakes and ponds freeze only at the top layer and at the bottom it does not, but if the water freezes at the bottom also then animal and plant life would not be possible.

The anomalous behavior of water, sometimes called the density anomaly, is due to strong intermolecular attractions between water molecules called hydrogen bonds. The large electronegativity difference between oxygen and hydrogen causes the hydrogen-oxygen bonds to be polar.

Specific heat capacity

Specific heat, the quantity of heat required to raise the temperature of one gram of a substance by one Celsius degree. The units of specific heat are usually calories or joules per gram per Celsius degree. For example, the specific heat of water is 1 calorie (or 4.186 joules) per gram per Celsius degree.

But let's try to understand the specific heat in detail.

Suppose you have 100 g of water in a vessel at 20 C temperature and you put that vessel on top of a stove (source of heat).  Place the thermometer inside it and hold a stopwatch. The heat from the stove will heat up the water and will raise its temperature that can be seen on the thermometer.

1. First note the time for increasing the temperature of the water from 20 degrees Celsius to 40 degrees Celsius (rise of 20 C).
2. Now you have water at 40 C, now again note the time for increasing the temperature of water upto 60 C (a rise of 40 C). You will notice that it takes double the time if we double the rise of temperature.

From the above experiment, we have the following conclusion.

Heat required to raise the temperature of a substance is proportional to the rise in temperature

Now we will do this experiment again but now with double quality. We now have 200 g of water at 20 C in a vessel which is kept on a stove and keeping all other things the same.

3. If we now try to raise the temperature of it upto 40 C and note the time of it.  We will notice that you need double the time as we got when we had only 100 g of water. (case 1)

Thus, Heat required to raise the temperature of the substance is proportional to the amount of substance.

Now in the next experiment, we change the liquid from water to oil. We take 100 g of oil at 20 C in a vessel keeping all other things the same.

1. If we now try to raise its temperature upto 40 C and note the time of it. You will get the time that is very less as compared to we get in case 1

Thus, the Heat required to raise the temperature of the substance depends on the nature of the substance.

The above series of experiments shows that the quantity of heat required to warm a given substance depends on its mass, m, the change in temperature, ΔT and the nature of the substance. The change in temperature of a substance, when a given quantity of heat is absorbed or rejected by it, is characterized by a quantity called the heat capacity of that substance.  Every substance has a fixed value of heat capacity.

Now you will understand the definition of specific heat more clearly.

Specific heat is defined as the amount of heat per unit mass absorbed or rejected by the substance to change its temperature by one.

Molar heat capacity

Heat capacity per mole of the substance is defined as the amount of heat (in moles) absorbed or rejected (instead of mass m in kg) by the substance to change its temperature by one unit.

Mathematically ,   q = n Cm ΔT   ;  Cm= q / (n ΔT)

Here q= heat absorbed in Joules (J), n= number of moles,

C_m = molar specific heat expressed in J mol -1K^-1 or J mol-1C^-1.

Heat transfer can be achieved by keeping either pressure or volume constant, accordingly, we define Cv and Cp. Let us discuss this now.

Molar-specific heat capacity at constant volume (Cv)

If the volume of the gas is maintained during the heat transfer, then the corresponding molar-specific heat capacity is called molar-specific heat capacity at constant volume (Cv).

Water has the highest specific heat of capacity because of which it is used as a coolant in automobile radiators and in hot water bags.

Molar-specific heat capacity at constant pressure (Cp)

If the gas is held under constant pressure during the heat transfer, then the corresponding molar-specific heat capacity is called molar-specific heat capacity at constant pressure (Cp).

A fun thing to do: Virtual lab

Below is the link of the states of matter simulations.

States of matter

What can we do in this simulation?

• We can choose from the options of gases available ( Neon, Argon, Oxygen and water)
• Then we can choose the state of matter from solid, liquid and gas.

We can then observe the temperature at which these gases are in different states. Like if we choose Water and Gas then the temperature would be around 373K or more, which says that water will be in gaseous form at 373 K and above.

• We can also choose one gas and then raises or lower its temperature and with the help of the motion of its atoms or molecules we can say whether it is in liquid , gaseous or solid states

Introduction

As all molecules of a gas are in a state of rapid and continuous motion, various properties of a gas like pressure, temperature, energy are explained and the kinetic theory was developed by Scottish physicist James Maxwell and Austrian Physicist Ludwig Boltzmann.

Kinetic theory of gases is based on the molecular picture of matter. It correlates the macroscopic properties (like pressure and temperature) of gases to microscopic properties like speed, kinetic energy of gas molecules.

Kinetic theory explains the behaviour of gases based on the idea that the gas consists of rapidly moving atoms or molecules. This is possible as the interatomic forces, which are short range forces that are important for solids and liquids, can be neglected for gases.

Dalton’s Atomic Theory

• Atomic hypothesis was given by many scientists. According to which everything in this universe is made up of atoms.
• Atoms are little particles that move around in a perpetual order attracting each other when they are little distance apart.
• But if they are forced very close to each other then they rebel.

For example: - Consider a block of gold. It consists of molecules that are constantly moving.

• Dalton’s atomic theory is also referred to as the molecular theory of matter. This theory proves that matter is made up of molecules which in turn are made up of atoms.

According to Gay Lussac’s law - when gases combine chemically to yield another gas, their volumes are in ratios of small integers.

Avogadro’s law states that the equal volumes of all gases at equal temperature and pressure have the same number of molecules.

Conclusion: - All these laws proved the molecular nature of gases.

Dalton’s molecular theory forms the basis of Kinetic theory.

Behaviour of Gases

Gases at low pressures and high temperatures much above that at which they liquefy (or solidify) approximately satisfy a relation between their pressure, temperature and volume:

P V= K T . This is the universal relation which is satisfied by all gases. Where P, V, T are pressure, volume and temperature resp. and K is the constant for a given volume of gas. It varies with the volume of gas. K=NK_B

Where N=number of molecules and k_B = Boltzmann Constant and its value never change.

From above two equations we have

P V=NK_BT⇒  PV/NT=K_B Which is the same for all gases.

Consider there are 2 gases :- (P1, V,1, T1) and (P2, V2, T2) where P, V and T are pressure, volume and temperature respectively.

This is Avogadro’s hypothesis, that the number of molecules per unit volume is same for all gases at a fixed temperature and pressure

Conclusion: - This relation is satisfied by all gases at low pressure and high temperature.

According to Avogadro’s hypothesis, the number of molecules per unit volume is the same for all gases at a fixed P and T.

Avogadro number is denoted by NA. Where A denotes Avogadro number.

NA= 6.022 ×10²³. It is a universal value.

Experimentally it has been found that the mass of 22.4 litres of any gas is equal to molecular weight in grams at standard temperature and pressure.

Perfect Gas Equation

Perfect gas equation is given by PV=μRT,

Where P, V are pressure, volume, T =absolute temperature, μ = number of moles and R =universal gas constant.

R= kBNA where kB = Boltzmann constant and NA = Avogadro’s number

This equation tells about the behaviour of gas in a particular situation. If a gas satisfies this equation then the gas is known as Perfect gas or an ideal gas.

Ideal gas: A gas that satisfies the perfect gas equation exactly at all pressures and temperatures.  Ideal gas is a theoretical concept.

• No real gas is truly ideal. A gas which is ideal is known as real gas.
• Real gases approach the ideal gas behaviour for low pressures and high temperatures

Dalton’s Law of partial pressures

Dalton’s law of partial pressure states that the total pressure of a mixture of ideal gases is the sum of partial pressures.

Consider if there are several ideal gases mixed together in a vessel, then the total pressure of that vessel is equal to the sum of partial pressure.

Partial pressure is the pressure exerted by a particular gas if only that gas is present in the vessel.

For example: - Consider if in a vessel there is a mixture of 3 gases, A,B and C.So the partial pressure of A is equal to pressure exerted only by A and considering B and C are not present.

Similarly partial pressure of B is equal to the pressure exerted only by B and considering A and C are not there.  And Similarly for C.

According to Dalton’s law the total pressure of mixture is sum of partial pressure of A, partial pressure of B and partial pressure of C

Therefore P= P1+P2+---total pressure due to the mixture of gases is equal to the sum of the partial pressure of the gas.

Kinetic Theory of an Ideal Gas

Basis and assumptions of Kinetic Theory: -

1. Molecules of gas are in incessant random motion, colliding against one another and with the walls of the container.

2. All collisions are elastic. And total Kinetic energy and momentum are conserved.  In case of an elastic collision total Kinetic energy and momentum before collision is equal to the total Kinetic energy and momentum after collision.
3. The density and the distribution of the molecules is uniform throughout the gas.
4. Between two collisions a molecule moves in a straight path with uniform velocity. But when they come closer they experience the intermolecular forces and as a result their velocities change.
5. There are no intermolecular forces between the molecules of gas except during collisions.
6. There will be no force ,between the molecules. As a result molecules are moving freely as per Newton's first law of motion.
7. At ordinary temperature and pressure the molecular size is very small as compared to intermolecular distance between them.

In the above pictures we can see that molecules moving randomly first and then molecules colliding with each other and change their direct.

The Pressure of an Ideal Gas Based on Kinetic Theory

• Consider a container in the shape of a cube that is filled with an ideal gas. Only one molecule will be considered; the molecule collides with the container's walls and bounces back.
• Let the molecule's velocity while moving be (v_x, v_y, v_z).
• The velocity of the molecule as it bounces back will be (-v_x, v_y, v_z).
• The change in momentum = P_f – P_i where P_f = final momentum and P_i = initial momentum)
•   P_f-P_i= =mv_x-mv_x= -2 mv_x 
• The wall receives this change in momentum as a result of the contact.
• One molecule's momentum delivered to the wall in a collision= 2mv_x

However, because there are so many molecules, we must calculate the overall momentum transferred to the wall by them all.

To figure out how many molecules hit the wall, do the following:

• The area of the wall will be ‘A’. Therefore, in time Δt within a distance of  AvxΔt  all the molecules can hit the wall.
• If n be the number of molecules per unit volume and on average, half of the molecules will hit the wall and half of them will move away from the wall. Therefore,  will hit the wall.

• The total momentum will be:.
• The force exerted on the wall is equal to the rate of change of momentum which will be equal to 
• The Pressure on the wall is equal to  This is true for molecules having velocity v_x

Note:

• The velocity of all the molecules in the gas will not be the same. The velocities of each will be different.
• As a result, the following equation is valid for pressure due to a group of molecules moving at v_x in the x-direction, where n is the number density of that group of molecules.

As a result, the total pressure owing to all such groups may be calculated by adding the contributions due to each molecule. 

• Because the gas is isotropic, the molecules travel at random, meaning that their velocity can be in any direction.
• 
• Therefore, the pressure is equal to   where v2 is the average square speed.

Kinetic Interpretation of Temperature

A molecule's average kinetic energy is proportional to the absolute temperature of the gas. It is unaffected by the ideal gas's pressure, volume, or nature.

For the equation:  , by multiplying both sides by V we will get

   

Also  nV= N ( total number of molecules)

After simplifying the above equation 

Here

‘N’ is the number of molecules in a sample.

EN=12mv2=32kBT ......equation (3)

Therefore, the above equation depicts the average kinetic energy.

So, kinetic energy is directly proportional to the temperature. So, temperature can be identified as a molecular quantity.

Kinetic Theory: Consistent With Ideal Gas Equation and Gas Laws

1. It is consistent with the ideal gas equation:

For the kinetic gas equation: 

For an ideal gas, its internal energy is directly proportional to the temperature. This depicts that internal energy of an ideal gas is only dependent on its temperature, not on pressure or volume.

2. When Kinetic theory is consistent with Dalton’s Law of partial pressure:

The equation for Kinetic theory  

If the mixture of gases is present in the vessel then 

The average Kinetic energy of the molecules of different gases at equilibrium will be equal   

Then the total pressure P is given by

So we have Total pressure P of mixture of gases as equal to the sum of partial pressure of individual gasesP= P₁ + P₂+ P₃............ This is known as Dalton’s law of partial pressure.