Internal Energy, First Law of Thermodynamics and Specific Heat Capacities of a Gas

Internal Energy(U)

Energy contained in a system is called internal energy. Total energy of a system is sum of internal kinetic energy and internal potential energy.

$$\text {Total Energy(E)} = \text {Internal Kinetic Energy (KE)} +\text {Internal Potential Energy (PE)}$$

$$E = KE + PE$$

$$\text {For ideal gas, E} = KE $$

Internal KE depends upon a velocity of molecules which depends upon KE i.e. temperature and internal potential energy depends on the position of a molecule in the system.

In the case of ideal gas, there is no force of attraction in molecules i.e internal PE is zero so, the internal energy of the ideal gas is only due to internal kinetic energy therefore in the case of the ideal gas internal energy depends on upon temperature only.

First Law of Thermodynamics

First law of thermodynamics states that "If the certain quantity is supplied to a system capable of doing work amount of heat absorbed by the sum of increase in internal energy of system and work done by the system."

If dQ is the amount of heat observed by the system and dV is the increase in internal energy of system then dW is the work done by the system according to the firs law of thermodynamics.

$$dQ = dU + dW\dots (i)$$

$$dQ = dU + PdV\dots (ii)$$

Above equations are the mathematical forms of the first law of thermodynamics.

Specific Heat Capacities of a Gas

If the amount of heat required to raise the temperature of unit mass of a substance to raise 1-degree centigrade.

When solid or liquid is heated there is a small change in volume and pressure so this can be neglected but when gas is heated there is a large change in volume and pressure of gas. A large amount of heat may be used in work done by gas. So, the specific heat capacity of gas without imposing the condition (const V or P) is meaningless so the gas has two specific heat capacity of the gas at constant volume and specific heat capacity at a constant temperature.

Specific heat capacity at constant volume

It is defined as the quantity of heat required to raise the temperature of 1 kg of a gas through 1 K at constant volume. It is denoted by c_v and its SI-unit is Jkg^-1K^-1.

Molar heat at constant volume

It is defined as the amount of heat required to raise the temperature of 1 mole of a gas through 1 kelvin at constant volume. It is denoted by c_v and its SI-unit is Jkg^-1K^-1.

If Q is an amount of heat required to raise the temperature of n-mole of a gas through \(\Delta T\) at constant volume, the molar heat capacity of the gas is

$$C_v = \frac {Q}{n\Delta T}$$

If molar mass of the gas is M, then

$$C_v = M c_v$$

Specific heat capacity at constant pressure

It is defined as the quantity of heat required to raise the temperature of 1 kg of a gas through 1 K at constant pressure. It is denoted by c_pand its SI-unit is Jkg^-1K^-1.

Molar heat at constant pressure

It is defined as the amount of heat required to raise the temperature of 1 mole of a gas through 1 kelvin at constant pressure. It is denoted by c_pand its SI-unit is Jkg^-1K^-1.

If Q is an amount of heat required to raise the temperature of n-mole of a gas through \(\Delta T\) at constant pressure, the molar heat capacity of the gas is

$$C_p = \frac {Q}{n\Delta T}$$

If molar mass of the gas is M, then

$$C_p = M c_p$$

C_p is greater than C_v

At constant volume, all heat energy supplied to it is used to increase the internal energy only and the temperature of a gas. But in the case of constant pressure and here total heat energy supplied is used to increase the internal energy as well as work against external pressure. So, more heat will be required for increasing the temperature of the gas when gas is heated to the same temperature. Hence, the specific heat of a gas at constant pressure is greater than the specific heat at constant volume, C_p > C_v.

Relation between Two Specific Heat Capacities of a Gas

Consider n-mole of an ideal gas taken in a cylinder, fitted with a weightless, frictionless and moveable piston. Let P, V, and T be the initial pressure, volume and temperature of the gas.

Suppose the gas is heated at constant volume, and its temperature increases by dT, then the amount of heat supplied to the system

$$dQ = n C_v \times dT \dots (i)$$

where C_v is the molar specific heat capacity at constant volume.

As the gas is heated at constant volume, it will not perform external work. Therefore, from the first law of thermodynamics, the amount of heat supplied, dQ will be equal to increase in internal energy of a gas.

$$dQ = dU + PdV = dU + 0 = dU$$

$$dU = n C_v \times dT \dots (ii)$$

Returning to the initial condition, the gas is heated at constant pressure and its temperature increases by same amount dT. If dQ is the amount of heat supplied, then

$$dQ = n C_p \times dT \dots (iii)$$

where C_p is the molar specific heat capacity at constant pressure. As the heat supplied is usedin two processes: (i) to increases the internal energy, dU and (ii) to do external work, dW. Then, from the first law of thermodynamics, we have

$$dQ = dU + PdV \dots (iv)$$

From equation (ii)c and (iv), we have

$$n C_p \times dT =dU + PdV $$

Since the rise in internal energy is independent of volume of the gas, and only depends on temperature, equation (v) can be written as

$$n C_p \times dT = n C_v \times dT + PdV \dots (v)$$

For n-mole of an ideal gas we have

$$PV = n RT \dots (vi)$$

differentiating equation (vi) with respect to T at constant Pressure we get

$$\frac {d}{dT}PV = \frac {d}{dT} (n RT) $$

$$P\frac{dV}{dT} = nR\frac{dT}{dT}$$

$$PdV = nR dT \dots (vii) $$

from equation(v) and (vii)

$$n C_p \times dT = n C_v \times dT + nR dT$$

$$C_p = C_v + R $$

$$C_p - C_v = R $$

This formula is known as the Meyer's formula.

If M is the molar mass of the gas, then equation (vii) can be written as

$$\frac {C_p}{M} - \frac {C_v}{M} = \frac {R}{M}$$

$$c_p - c_v = r$$

where r is the gas constant per unit mass.