Physics THERMAL EQUILIBRIUM, ZEROTH LAW OF THERMODYNAMICS, INTERNAL ENERGY AND FIRST LAW OF THERMODYNAMICS

### Topic Covered

INTRODUCTION
THERMAL EQUILIBRIUM
ZEROTH LAW OF THERMODYNAMICS
HEAT & INTERNAL ENERGY AND WORK
FIRST LAW OF THERMODYNAMICS

### INTRODUCTION

So far, we have studied thermal properties of matter. In this chapter we shall study laws that govern thermal energy. We shall study the processes where work is converted into heat and vice versa.

In winter, when we rub our palms together, we feel warmer; here work done in rubbing produces the ‘heat’. Conversely, in a steam engine, the ‘heat’ of the steam is used to do useful work in moving the pistons, which in turn rotate the wheels of the train.

color{blue} • Thermodynamics is the branch of physics that deals with the concepts of heat and temperature and the inter-conversion of heat and other forms of energy. Thermodynamics is a macroscopic science. It deals with bulk systems and does not go into the molecular constitution of matter. In fact, its concepts and laws were formulated in the nineteenth century before the molecular picture of matter was firmly established.

color{blue} • Thermodynamic description involves relatively few macroscopic variables of the system, which are suggested by common sense and can be usually measured directly.

A microscopic description of a gas, for example, would involve specifying the co-ordinates and velocities of the huge number of molecules constituting the gas. The description in kinetic theory of gases is not so detailed but it does involve molecular distribution of velocities.

color{blue} • Thermodynamic description of a gas, on the other hand, avoids the molecular description altogether. Instead, the state of a gas in thermodynamics is specified by macroscopic variables such as pressure, volume, temperature, mass and composition that are felt by our sense perceptions and are measurable*.

color{blue} • 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.

color{blue} • Thermodynamics is not concerned with the motion of the system as a whole. It is concerned with the internal macroscopic state of the body. When a bullet is fired from a gun, what changes is the mechanical state of the bullet (its kinetic energy, in particular), not its temperature.

When the bullet pierces a wood and stops, the kinetic energy of the bullet gets converted into heat, changing the temperature of the bullet and the surrounding layers of wood. Temperature is related to the energy of the internal (disordered) motion of the bullet, not to the motion of the bullet as a whole.

### 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 characterise 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.

In general, whether or not a system is in a state of equilibrium depends on the surroundings and the nature of the wall that separates the system from the surroundings. Consider two gases A and B occupying two different containers.

We know experimentally that pressure and volume of a given mass of gas can be chosen to be its two independent variables.

Let the pressure and volume of the gases be (P_A, V_A) and (P_B, V_B) respectively. Suppose first that the two systems are put in proximity but are separated by an color{gray} "adiabatic wall" – an insulating wall (can be movable) that does not allow flow of energy (heat) from one to another.

The systems are insulated from the rest of the surroundings also by similar adiabatic walls. The situation is shown schematically in Fig. 12.1 (a). In this case, it is found that any possible pair of values (P_A, V_A) will be in equilibrium with any possible pair of values (P_B, V_B).

Next, suppose that the adiabatic wall is replaced by a color{orange} "diathermic wall" – a conducting wall that allows energy flow (heat) from one to another. It is then found that the macroscopic variables of the systems A and B change spontaneously until both the systems attain equilibrium states. After that there is no change in their states.

The situation is shown in Fig. 12.1(b). The pressure and volume variables of the two gases change to (P_B ′, V_B ′) and (P_A ′, V_A ′) such that the new states of A and B are in equilibrium with each other**. There is no more energy flow from one to another. We then say that the system A is in thermal equilibrium with the system B.

In thermal equilibrium, the temperatures of the two systems are equal. We shall see how does one arrive at the concept of temperature in thermodynamics? The Zeroth law of thermodynamics provides the clue.

### ZERO^(TH) LAW OF THERMODYNAMICS

Imagine two systems A and B, separated by an adiabatic wall, while each is in contact with a third system C, via a conducting wall [Fig. 12.2(a)].

The states of the systems (i.e., their macroscopic variables) will change until both A and B come to thermal equilibrium with C. After this is achieved, suppose that the adiabatic wall between A and B is replaced by a conducting wall and C is insulated from A and B by an adiabatic wall [Fig.12.2(b)].

It is found that the states of A and B change no further i.e. they are found color{green} "to be in thermal equilibrium with each other".

This observation forms the basis of the color{brown} "Zeroth Law of Thermodynamics", which states that color{gray} "two systems in thermal equilibrium with"  color{gray}"a third system separately are in thermal equilibrium with each other’". R.H. Fowler formulated this law in 1931 long after the first and second Laws of thermodynamics were stated and so numbered.

The Zeroth Law clearly suggests that when two systems A and B, are in thermal equilibrium, there must be a physical quantity that has the same value for both.

This thermodynamic variable whose value is equal for two systems in thermal equilibrium is called temperature (T ). Thus, if A and B are separately in equilibrium with C, T_A = T_C and T_B = T_C. This implies that T_A = T_B i.e. the systems A and B are also in thermal equilibrium.

We have arrived at the concept of temperature formally via the Zeroth Law.

### HEAT, INTERNAL ENERGY AND WORK

The Zeroth Law of Thermodynamics led us to the concept of temperature that agrees with our commonsense notion. Temperature is a marker of the ‘hotness’ of a body.

It determines the direction of flow of heat when two bodies are placed in thermal contact. Heat flows from the body at a higher temperature to the one at lower temperature.

The flow stops when the temperatures equalise; the two bodies are then in thermal equilibrium. We saw in some detail how to construct temperature scales to assign temperatures to different bodies. We now describe the concepts of heat and other relevant quantities like internal energy and work.

The concept of internal energy of a system is not difficult to understand. We know that every bulk system consists of a large number of molecules. Internal energy is simply the sum of the kinetic energies and potential energies of these molecules.

We remarked earlier that in thermodynamics, the kinetic energy of the system, as a whole, is not relevant. Internal energy is thus, the sum of molecular kinetic and potential energies in the frame of reference relative to which the centre of mass of the system is at rest.

Thus, it includes only the (disordered) energy associated with the random motion of molecules of the system. We denote the internal energy of a system by U.

Though we have invoked the molecular picture to understand the meaning of internal energy, as far as thermodynamics is concerned, U is simply a macroscopic variable of the system.

The important thing about internal energy is that it depends only on the state of the system, not on how that state was achieved. Internal energy U of a system is an example of a thermodynamic ‘state variable’ – its value depends only on the given state of the system, not on history i.e. not on the ‘path’ taken to arrive at that state.

Thus, the internal energy of a given mass of gas depends on its state described by specific values of pressure, volume and temperature. It does not depend on how this state of the gas came about. Pressure, volume, temperature, and internal energy are thermodynamic state variables of the system (gas).

If we neglect the small intermolecular forces in a gas, the internal energy of a gas is just the sum of kinetic energies associated with various random motions of its molecules. We will see in the next chapter that in a gas this motion is not only translational (i.e. motion from one point to another in the volume of the container); it also includes rotational and vibrational motion of the molecules (Fig. 12.3).

Let's discuss the ways of changing internal energy of a system,Consider again, for simplicity, the system to be a certain mass of gas contained in a cylinder with a movable piston as shown in Fig. 12.4.

Experience shows there are two ways of changing the state of the gas (and hence its internal energy). 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.

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. Of course, 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. In short, heat and work are two different modes of altering the state of a thermodynamic system and changing its internal energy.

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. This is not just a play of words. The distinction is of basic significance.

The state of a thermodynamic system is characterised 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.

To summarise, heat and work in thermodynamics are not state variables. They are modes of energy transfer to a system resulting in change in its internal energy, which, as already mentioned, is a state variable.

In ordinary language, we often confuse heat with internal energy. The distinction between them is sometimes ignored in elementary physics books. For proper understanding of thermodynamics, however, the distinction is crucial.

### FIRST LAW OF THERMODYNAMICS

We have seen that the internal energy U of a system can change through two modes of energy transfer : heat and work. Let

color{orange} "ΔQ =" Heat supplied to the system by the surroundings
color{orange} "ΔW =" Work done by the system on the surroundings
color{orange} "ΔU =" Change in internal energy of the system

The general principle of conservation of energy then implies that

color{purple} "ΔQ = ΔU + ΔW"

...............12.1

i.e. the energy (ΔQ) supplied to the system goes in partly to increase the internal energy of the system (ΔU) and the rest in work on the environment (ΔW). Equation (12.1) is known as the color{lime} "First Law of Thermodynamics".

It is simply the general law of conservation of energy applied to any system in which the energy transfer from or to the surroundings is taken into account.

Let us put Eq. (12.1) in the alternative form

color{purple} " ΔQ – ΔW = ΔU"

........... (12.2)

Now, the system may go from an initial state to the final state in a number of ways. For example, to change the state of a gas from (P_1, V_1) to (P_2, V_2), we can first change the volume of the gas from V_1 to V_2, keeping its pressure constant i.e. we can first go the state (P_1, V_2) and then change the pressure of the gas from P_1 to P_2, keeping volume constant, to take the gas to (P_2, V_2).

Alternatively, we can first keep the volume constant and then keep the pressure constant. Since U is a state variable, ΔU depends only on the initial and final states and not on the path taken by the gas to go from one to the other.

However, ΔQ and ΔW will, in general, depend on the path taken to go from the initial to final states. From the First Law of Thermodynamics, Eq. (12.2), it is clear that the combination color{lime} "ΔQ – ΔW", is however, path independent.

This shows that if a system is taken through a process in which ΔU = 0 (for example, isothermal expansion of an ideal gas),

color{purple} "ΔQ = ΔW"

i.e., heat supplied to the system is used up entirely by the system in doing work on the environment.

If the system is a gas in a cylinder with a movable piston, the gas in moving the piston does work. Since force is pressure times area, and area times displacement is volume, work done by the system against a constant pressure P is

color{purple} "ΔW = P ΔV"

where ΔV is the change in volume of the gas.

Thus, for this case, Eq. (12.1) gives

color{purple} "ΔQ = ΔU + P ΔV" ..................... (12.3)

As an application of Eq. (12.3), consider the change in internal energy for 1 g of water when we go from its liquid to vapour phase. The measured latent heat of water is 2256  J/g. i.e., for 1 g of water ΔQ = 2256 J. At atmospheric pressure, 1 g of water has a volume 1 cm^3 in liquid phase and 1671 cm^3 in vapour phase.

Therefore, color{orange} {ΔW =P (V_g –V_l ) = 1.013 ×10^5 ×(1670)×10^-6 =169.2 J}

Equation (12.3) then gives

ΔU = 2256 – 169.2 = 2086.8 J

We see that most of the heat goes to increase the internal energy of water in transition from the liquid to the vapour phase.