Thermal Equilibrium

THERMAL EQUILIBRIUM AND TEMPERATURE

Two systems are said to be in thermal equilibrium with each other, if they are at the same temperature.

ZEROTH LAW OF THERMODYNAMICS

It states that if two systems A and B are in thermal equilibrium with a third system C, then A and B must be in thermal equilibrium with each other.

Fig. 1.01 shows two systems A and B separated by an adiabatic wall (a wall which does not allow hear flow).The two systems are placed in contact with a third system C with a diathermic wall (a wall which permits heat flow) in between.

Suppose that the system A, B and C are at different temperatures. Obviously, the three systems will not be in thermal equilibrium with one another. However, the systems A and C; and the systems B and C will exchange heat and after certain time, they will attain thermal equilibrium separately. If the adiabatic wall between the systems A and B is removed at that time, it will be found that there is no exchange of heat between the systems A and B. Therefore, the systems A and B also acquire thermal equilibrium, when the systems A and B are allowed to attain thermal equilibrium separately with the system C.

i) Thermodynamic system: An assembly of extremely large number of particles having a certain value of pressure, volume and temperature is called a thermodynamic system. For example, a large collection of gas molecules is a thermodynamic system. ii) Thermodynamic variables: The variables which determine the thermodynamic behaviour of a system are called thermodynamic variables. The quantities like pressure (P), volume (V) and temperature (T) are thermodynamic variables. There are some other thermodynamic variables, such as internal energy (U), entropy (S), etc. All other thermodynamic variables can be expressed in terms of P,V and T. iii) Equation of state: A relation between pressure, volume and temperature for a system is called its equation of state. The state of a system is completely known in terms of its pressure, volume and temperature. For example, for I mole of an ideal gas, the equation of state is PV=RT

iv) Thermodynamic process: A thermodynamic process is said to be taking place, if the thermodynamic variables of the system change with time. In practice, the following types of thermodynamic processes can take place. a) Isothermal process: A thermodynamic process that takes place at constant temperature is called isothermal process. b) Isobaric process: A thermodynamic process that takes place at constant pressure is called isobaric process. c) Isochoric process: A thermodynamic process that takes place at constant volume is called isochoric process. d) Adiabatic process: A thermodynamic process in which no heat enters or leaves the system is called adiabatic process. e) Cyclic process: A thermodynamic process in which the system returns to its original state is called a cyclic process.

1.04 INTERNAL ENERGY OF A GAS Intermolecular potential energy of a real gas is function of its volume. The molecules of a gas are always in random motion and according to the kinetic theory of gases; the average kinetic energy of the gas molecule is given by

Therefore, the kinetic energy of a gas is function of the temperature of the gas. The internal energy of a gas is defined as the sum of the kinetic energy and the intermolecular potential energy of the molecules of the gas. A graphical representation of the state of a system with the help of two thermodynamic variables is called indicator diagram of the system.

Importance of P-V diagram: It can be mathematically proved that the work done by a system or on the system is numerically equal to the area under the P-V diagram.
Since work done by a system is taken as positive and work done on the system as negative, it follows that if area under P-V diagram is traced in clockwise direction, then work done will be positive and it will be negative, if the area is traced in anticlockwise direction.
Consider a gas enclosed in a cylinder provided with a frictionless and weightless piston. Suppose that the gas undergoes expansion from the initial state is corresponding to pressure P1 and volume V1 to the final state B, when the pressure and volume become P2 and V2 respectively. Let us calculate the work done by the gas.
Analytical method: Suppose that at any instant during the expansion, the pressure and volume of the gas are P and V respectively (Fig.1.04). Further, suppose that the piston moves through infinitesimally small distance dx again the constant pressure P, so that volume becomes V+ dV. Then, small work done.

dW= force on piston x small distance moved.
If a is area of cross-section of the piston, then the force on piston will be P a. therefore, dW=Pa x dx Now a x dx =dV (the small increase in volume of the gas) dW= P dV

The total work W done during the expansion of the gas from the initial state A (P1, V1) to the final state B (P2, V2) can be obtained by integrating the equation (1.02) between the limits V1 to V2,)
Therefore, total work done is given by

As already said, during expansion, work is done by the gas and hence it is positive. In case of compression, work is done on the gas and it is negative.

1.07. WORK DONE DURING A CYCLIC PROCESS Suppose that the gas enclosed in a cylinder is expanded from the initial state A to the final state B along the path AXB (Fig. 1.06). Let W1 be the work done during expansion. Then, W1= area AXBbaA

During expansion of the gas, work is done by the system, which according to the sign convention has been taken as positive. Now amount of work done along the cyclic path AXBYA, W=W1 + W2 = (area AXBbaA) + (-area BYAabB) = + area AXBYA
Thus, net amount of work done along a cyclic path in numerically equal to the area of the cyclic path. For the cyclic path shown in Fig. 1.06, as the area comes out to be positive, net work will be cone by the system. It may be noted that the cyclic path has been traced in clockwise direction and expansion curve AXB lies above the compression curve BYA.

Conclusions: It follows that in a cyclic process:

i) If the cyclic path is traced in clockwise direction or the expansion curve lies above the compression curve, the work will be cone by the system and ii) If the cyclic path is traced in anticlockwise direction or the expansion curve lies below the compression curve, the work will be done on the system.

1.08FIRST LOW OF THERMODYNAMICS First law of thermodynamics is, in fact, the law of conservation of energy. According to the law of conservation of energy, the energy can neither be created nor it can be destroyed but can change itself from one form to another. According to first law of thermodynamics, if an amount of heat dQ is added to a system, a part of it may increase its internal energy by an amount dU, while the remaining part may be used up as the external work dW done by the system. Thus, if dQ, dU and dW all are in same units, then dQ = dU + dW Consider a gas enclosed in a cylinder provided with a frictionless and weightless piston. Suppose that corresponding to the initial state A, pressure and volume of the gas are P and V respectively. Further, suppose that the piston moves through an infinitesimally small distance dx at constant pressure P(Fig.1.07), so that its volume in the final state B becomes V+dV. Then, the small work done, dW= force on piston x dx

If a is area of cross-section of the piston, then force on the piston will be equal to Pa. Therefore, dW=Pa x dx =P x (a dx) Now a x dx = dV, the small increase in volume. Therefore, dW = P dV Hence, the equation (1.06) may be rewritten as dQ =dU + PdV The equations (1.06) and (1.07) are mathematical forms of the first law of thermodynamics. The signs of dQ, dU and dW are known from the following sign conventions: Sign conventions: 1. Work done by a system is taken as positive, while work done on the system is taken as negative. 2. Heat gained (added) by a system is taken as positive, while the heat lost (extracted) by the system is taken as negative. 3. The increase in internal energy of system is a taken as positive, while the decrease in internal energy is taken as negative.

1.09. APPLICATIONS OF THE FIRSTLAW OF THERMODYNAMICS TO A CYCLIC PROCESS AND BOILING PROCESS 1. Cyclic process: In a cyclic process, the system returns to its initial state at the end of the cycle. Fig. 1.08 shows a cyclic process, in which the system proceeds from the state A to B along the path X and then returns back to the initial state A along the path Y. Obviously, the change in internal energy of the system along the complete cyclic process is zero i.e. dU =0. Therefore, for a cyclic process, the first law of thermodynamic becomes dQ = 0+ P dV dW = P dV in a cyclic process, the heat supplied to the gas is wholly converted into work and the area of the cyclic process is numerically equal to the work done during the process. 2. Boiling process: When a liquid is supplied heat, it starts boiling at a particular temperature, called its boiling point. At boiling point (pressure kept constant), the unit mass of the liquid requires a definite amount of heat energy to change from liquid to vapour state. The amount of heat supplied is called latent heat of vaporization of the liquid. Consider a liquid of mass m at a temperature equal to its boiling point. Let P be the constant external pressure, at which it is boiling and L be its latent heat of vaporization. Then, Amount of heat required to convert whole of the liquid into vapour. AQ = m L From the first law of thermodynamics. Heat absorbed + increase in internal energy + external work performed. AQ = AU + A if U i and U f are the initial and final values of the internal energy, and V i and V f are initial and final values of the volume, then increase in internal energy, AU = U f - U i and external work done, AW = P dV = P (V f - V i) In the equation (1.08), substituting for AQ, AU and AW, we have m L = (U f - U i) + P (V f - V i) or U f - U I = m L- P(V f - V i) Knowing m, L, V f and V I the increase in internal energy can be calculated.

TWO SPECIFIC HEATS OF A GAS

The specific heat of a substance is define as the amount of heat required to raise the temperature of its unit mass to one degree centigrade (or Kelvin).

1. Specific heat of a gas at constant volume. It is defined as the amount of heat required to raise the temperature of 1g of a gas through 1oC at constant volume. It is denoted by cv.

Molar Specific heat at constant volume. It is defined as the amount of heat required to raise the temperature of 1 mole of a gas through 1oC at constant volume. It is denoted by Cv. If M is the molecular weight of the gas, then Cv = M cv 2. Specific heat of a gas at constant pressure. It is defined as the amount of heat required to raise the temperature of 1g of a gas through 10C at constant pressure. It is denoted by Cp.

Molar specific heat of a gas at constant pressure. It is defined as the amount of heat required to raise the temperature of 1 mole of a gas through 10C at constant pressure. It is denoted by Cp. Obviously, Cp=Mcp

Cp IS GREATER THAN Cv

When a gas is heated, both its volume and pressure change. Let us estimate the amount of heat required by the gas to heat it through 10C, when either its volume or its pressure is kept constant.

1. When volume is kept constant. The pressure on the gas is increased so that its volume remains constant. As the gas cannot perform work (V=constant), the heat supplied will increase only the temperature or the internal energy of the gas. Therefore, in case of Cv, heat is required only for increasing the temperature of the gas through 10C. 2. When pressure is kept constant. When the gas is heated at constant pressure, it expands also. Therefore, the heat supplied at constant pressure will partly increase its temperature (internal energy) and partly will be utilized in performing work against the external pressure. Therefore, in case of Cp, more heat (as compared to the case of Cv) will be required for increasing the temperature of the gas through 10C.

Hence, the specific heat of a gas at constant pressure is greater than the specific heat at constant volume. i.e. Cp>Cv. The difference between the values of two specific heats is equal to amount of heat equivalent to work performed by the gas during expansion at constant pressure.

RELATION BETWEEN TWO SPECIFIC HEATS OF GAS

The relation between the specific heat of a gas at constant volume and at constant pressure was first obtained by Robert Mayer in 1842.
Consider one mole of an ideal gas contained in a cylinder fitted with frictionless piston. Let A be the area of piston and P, V and T be pressure, volume and temperature of the gas respectively. Suppose that heat is supplied to the gas at a constant volume, till its temperature increases through dT. Then the amount of heat supplied.

dQ= 1*Cv*dT=Cv dT
Now, suppose that heat is supplied to the gas at constant pressure to again increases its temperature through dT. It dQ’ is the amount of heat supplied, then

dQ’=1*Cp*dT=CpdT
In fact, when the gas is heated at constant pressure, the piston moves outward and work (say dW) is performed by the gas. As the increase in temperature is the same (dT) in the two cases,

dQ’=dQ+dW/J ……(1.10)

Where dW/J is the heat equivalent of the work dW. If the piston moves out through a small distance, then dW=force * distance=(Pressure * area of the piston) *distance =(PA)dx = P dV,

Where dV(=A*dx)is the small increase in volume of the gas, when heated at constant pressure.
In equation (1.10), substituting for dQ, dQ’ and dW, we have

Cp dT= Cv dT + PdV/J
Or (Cp – Cv) dT= PdV/J ……(1.11)

According to the perfect gas equation PV=RT
The heat dQ’ is supplied to the gas at a constant pressure P. Therefore, differentiating both the sides of the above equation by treating P as constant, we have PdV=RdT
Substituting for PdV in the equation (1.11) we get