Chapter 12: Thermodynamics

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.

Thermodynamics processes: Quasi-static process

Quasi-static term means semi-static. It is not purely moving. It is a hypothetical construct which means it is not real. It is an infinitely slow process which means change from its original position is not at all significant.

System changes its variables (P, T, and V) so slowly that it remains in equilibrium with its surroundings throughout.

The characteristic for a system to be a Quasi-static process is that it is an extremely very slow process and there should not be any accelerated motion.

In a quasi-static process, the temperature of the surrounding reservoir and the external pressure differ only infinitesimally from the temperature and pressure of the system.

Isothermal process

When a thermodynamic system undergoes a process under the condition that its temperature remains constant, then the process is said to be an isothermal process. The essential condition for an isothermal process is that the system must be contained in a perfectly conducting chamber.

For the isothermal process, ΔU=0, since T= constant.    

From the first law of thermodynamics, Δ Q= Δ U+ΔW ⇒ ΔQ=ΔW.  Hence, for an ideal gas, all heat is converted into work in an isothermal process.

From Ideal gas equation  P V= nRT, for isothermal process T= constant    P V=constant   ;   P α  1/V  

Pressure and volume are inversely proportional to each other for isothermal processes.

Derivation of work done in the Isothermal process

Heat taken ΔQ  and work done is same in case of isothermal process

Adiabatic Process

When a thermodynamic system undergoes a process under the condition that no heat comes into or goes out of the system, then the process is said to be an adiabatic process. Such a process can occur when a system is perfectly insulated from its surroundings. These processes are sudden processes.

For the adiabatic process Q=0 (there is no heat transfer taking place).

From first law of thermodynamics  ΔQ=ΔU+ ΔW; since ΔQ=0

So we have Δ U= -ΔW  for adiabatic processes.

Conditions for adiabatic processes: The walls of the container should be perfectly non-conducting in order to prevent any exchange of heat between the gas and its surroundings. The process of compression or expansion must be rapid so that there is no time for the exchange of heat.

These conditions are ideal conditions and are difficult to achieve.

Equation of state of Adiabatic processes

The most widely used equation of state of adiabatic process is in terms of pressure and volume    P V^γ= constant

Here , where gamma is the ratio of specific heat at constant pressure and specific heat at constant volume. The value of gamma differs for monoatomic, diatomic and polyatomic gasses. We will discuss this in detail in the next chapter.

If we use the ideal gas equation  PV=nRT and PV^γ= constant

We will get two more equations of states for adiabatic processes

• In terms of volume and temperature:   T V^γ-1= constant
• In terms of pressure and temperature: Tγ P^1-γ= constant

Calculation of work done in case of adiabatic processes

Isobaric process

If the working substance is taken in an expanding chamber in which the pressure is kept constant, the process is called an isobaric process. In this process, the gas either expands or shrinks to maintain constant pressure and hence a net amount of work done by the system or on the system.

Δ P=0    V/T= constant

Isochoric process

If a substance undergoes a process in which the volume remains unchanged, the process is called an isochoric process. The increase of pressure and temperature produced by the heat supplied to a working substance contained in a non-expanding chamber is an example of the isochoric process.

Δ V=0   and  P/T= constant   

By first law of thermodynamics,   Δ Q= Δ U + P ΔV = ΔU

Cyclic Process

The process in which the initial and final state is the same is known as a cyclic process. It is a sequence of processes that leave the system in the same state in which it started. Hence, the work done by the system in a cyclic transformation is equal to the heat absorbed by the system.

In Cyclic Process, since the internal energy is a state variable, ΔU = 0, i.e., the internal change is zero. The initial and final internal energies remain equal.      

Therefore, the work done by the system in the cyclic process is equal to the heat that the system absorbs. In a P-V graph, where P is on the Y-axis and V is on the X-axis, the network involved in the cyclic process is the area enclosed in the diagram. If the cycle goes anticlockwise, then work is done on the system in every cycle.

Heat Engine

A cyclic device by which heat is converted into mechanical energy gets is called a heat engine. For a heat engine, there are essentially three elements:-

Source: A hot body at a fixed temperature T1 from which heat can be drawn heat is called a source or hot reservoir.

Sink:  A cold body which is at fixed temperature T2 to which any amount of heat can be rejected is called a sink or cold reservoir.

Working substance: The materials, on being supplied heat perform mechanical work is called working substance.

In a heat engine, the working substance takes in heat into the source, converts a part of it into external work, gives out the rest in the sink and returns to its initial state. This series of operations constitute a cycle.  The work can be continuously obtained by performing the same cycle and over again.

Suppose the working substance takes Q1 heat from the source at temperature T1 and gives out Q2 heat to the sink at temperature T2. Suppose W be the amount of work obtained. The net amount of heat absorbed by the substance Q1 -Q2, which has been actually converted into work.

According to the first law of thermodynamics for a full cycle,

Q1 - Q2 =W

Efficiency of the engine

The thermal efficiency of an engine is defined as the ratio of the work obtained to the heat taken in from the source that is  

This equation indicates that the efficiency of the heat engine will be unity when Q2=0, this is however not possible in practice. This means that the engine cannot convert all the heat taken from the source into work.

Refrigerator and heat pump

A refrigerator or heat pump is a device utilized for cooling things.

Key Elements

• A cold reservoir at temperature T2
• A working substance
• A hot reservoir at temperature T1

Working

• The working substance follows a cycle consisting of several processes.
• A sudden expansion of the gas from high to low pressure cools it and converts it into a vapor-liquid mixture.
• Absorption by the cold fluid of heat from the region to be cooled converts it into vapor.
• The vapor gets heated up due to external work done on the working substance.
• The heat gets released by the vapor to the surroundings bringing it to the initial state and completing the cycle.

Coefficient of performance of the refrigerator.

It is denoted by α . mathematically given as  α = Q2/W,

Where Q₂ is the heat extracted from the cold reservoir and 

W is the work done on the system- The refrigerant.

Also work done   . In a heat engine, heat cannot be fully converted to work; likewise, a refrigerator cannot work without some external work done on the system, i.e., the coefficient of performance cannot be infinite.

Second law of Thermodynamics

This has two statements. First is the kelvin-Planck statement which is based upon the performance of the heat engine and second is Clausius statement which is based on the performance of the refrigerator.

These are 2 statements of second law of thermodynamics given as,

Kelvin-Planck Statement: - No process is possible whose result is the absorption of heat from a reservoir and the complete conversion of the heat into work.

Clausius statement: - No process is possible whose result is the transfer of heat from a colder object to a hotter object.

Explanation of Kelvin-Planck Statement: It is always impossible that the total amount of heat which is supplied to the system will get converted to work, and there will always be loss of heat. Complete conversion of heat into work is not possible.

Explanation of Clausius statement: - Transfer of heat from a colder body to hotter body won’t take place until some external work is done on the system.

Reversible and Irreversible processes

Reversible Process

A thermodynamic process is reversible if the process can be turned back such that the system and surroundings return to their original states, with no other change anywhere else in the universe.

This means in the Reversible processes if a process starts from the initial state then it goes to the final state and then it can be reversed back from final state to the initial state.

Examples: - Isothermal expansion and compression, Electrolysis, extension of spring etc

A process is reversible if:-

• It is quasi-static
• No dissipative forces (that is no loss of heat by friction etc.).
• Both initial and final states of the system are in thermodynamic equilibrium with each other.

Irreversible Process

An irreversible process can be defined as a process in which the system and the surroundings do not return to their original condition once the process is initiated.   Irreversible processes are those that cannot be reversed.

Two causes which give rise to irreversible processes

• Irreversible processes take place at a very fast rate.
• Dissipative Effects.

Examples:-Plastic deformation, Combustion, Diffusion, Falling of water from the hill, relative motion with friction, heat transfer.

Carnot Engine

• A Carnot engine is named after a Carnot scientist.
• It is a reversible heat engine operating between two temperatures.
• It has a maximum efficiency which no other engine can have.

Construction

The essential sections of an ideal heat engine or Carnot heat engine are shown in the diagram below.

1. Source of heat: The source is maintained at a fixed higher temperature T1, from which the working substance takes heat. The source is supposed to possess the infinite thermal capacity and as such, any amount of heat can be taken from it without altering its temperature.
2. Sink of heat: The sink is maintained at a fixed lower temperature T2, to which any amount of heat can be emitted by the working substance. It also has the infinite thermal capacity and as such its temperature remains constant at T2, even when any amount of heat is emitted to it.
3. Working substance: A perfect gas plays the role of a working substance. It is contained in a cylinder with non-conducting sides but having a perfectly conducting base. This cylinder is fixed with a perfectly non-conducting and frictionless piston.

Apart from these essential parts, there is also a perfectly insulating stand or pad on which the cylinder can be positioned. It provides complete isolation to the working substance from the surroundings so that the gas can undergo adiabatic changes.

Cycle of processes in a Carnot engine

Basic Function of any heat engine is it will take heat Q1 from a hot reservoir at temperature T1 and give heat Q2   to a cold reservoir at temperature T2.

1. The system is absorbing heat so it is isothermal expansion. Engine absorbs heat Q1 at temperature T1.
2. An adiabatic expansion takes place inside the engine because of which there is increase in the temperature of the engine from T1 to T2 but no flow of heat.
3. The system is releasing heat so it is an isothermal contraction. Engine releases heat Q2 at temperature T2.
4. An adiabatic compression takes place again which changes the temperature of the system from T2 to T1.
5. One cycle of the Carnot engine will have Isothermal expansion then adiabatic process, and then isothermal contraction followed by an adiabatic process. This will keep on repeating.

The efficiency of Carnot engine is given by

Below is the P-V diagram describing every step of the cycle and the work done in each cycle?