Specific Heats of Gases: Formula, Constant Pressure, Ratio and Heat Capacity | Thermodynamics

Specific Heats of a Gas: Formula, Constant Pressure, Ratio and Heat Capacity!

Specific Heats of a Gas:

The specific heat capacity of a substance may be defined as the quantity of heat required to raise the temperature of unit mass of the substance by one degree. The unit of specific heat is J/kg-°C.

Since this unit is small it is convenient to express it in kJ/kg-K or kJ/kg-°C.

The quantity of heat required to raise the temperature of 1 kg of water by 1°C or 1 K is approximately 4.19 kj. Hence, the specific heat capacity of water is 4.19 kJ/kg-°C or 4.19 kJ/kg-K.

The solids and liquids have only one value of specific heat but a gas is considered to have two distinct values of specific heat capacity:

(i) A value when the gas is heated at constant volume, and

(ii) A value when the gas is heated at constant pressure.

Consider a quantity of a gas enclosed in a fixed container as shown in fig. 2-8. If heat is supplied to the gas the pressure and the temperature of the gas are raised in accordance with the formula-

P₁/T₁ = P₂/T₂ because the volume of the gas remains the same.

Thus, the whole of the heat energy is utilized in increasing the kinetic energy of the gas molecules, which will be indicated by the rise in the temperature. When the volume of the gas is constant the quantity of heat required to raise the unit mass of the gas to one degree C or K is known as the specific heat of the gas at constant volume and it is denoted by C_v.

Consider the gas to be enclosed in a cylinder fitted with a closely fitting but frictionless piston. This piston, carrying constant load W, is as shown in fig. 2-9.

When heat is supplied to a gas the kinetic energy of the gas molecules will increase but the tendency for the pressure to rise will be counter-balanced by the piston rising upward and thereby lifting the load and increasing the volume of the gas. Thus, the heat energy supplied is utilized in increasing the temperature and volume of the gas and in doing work.

The pressure remains constant throughout and is equal to the weight on the piston divided by the area of the piston. When the heat is supplied under these conditions, the quantity of heat required to heat the unit mass of the gas through one °C or K is known as the specific heat of the gas at constant pressure and it is denoted by C_p.

Thus, we see that the gases have two specific heats:

(i) At constant pressure, C_p, and

(ii) At constant volume, C_v.

The specific heat at constant pressure is greater than the specific heat at constant volume because heat supplied to a gas at constant pressure is utilized for two purposes:

(i) For increasing the internal energy, and

(ii) For doing the external work.

Ratio of Specific Heats:

The ratio (specific heat of a gas at constant pressure/specific heat of a gas at constant volume) is denoted Greek letter γ ‘gamma’. It is also called the adiabatic index. This ratio of specific heats is of importance in perfect gas calculations. According to the classical kinetic theory of gases this ratio should have the values 5/3, 7/5 and 4/7 respectively for monoatomic, diatomic and polyatomic gases.

This theory is not accurate; nevertheless at ordinary temperatures for monoatomic and diatomic gases the actual values of γ are not far from 1.67 and 1.4 respectively. For polyatomic gases there is a considerable deviation from the value 1.33. For air the value of γ is 1.4.

The properties of various real gases are given in table 2-2.

Relation between the Two Specific Heats of a Gas and the Specific Gas Constant:

Let T₁ be the initial temperature of a gas and T₂ be the final temperature of the gas.

Let V₁ and V₂ be the initial and final volumes of the gas and let P be the pressure of the gas; C_p and C_v be the specific heats of the gas at constant pressure and at constant volume respectively.

Both the specific heat capacities at constant pressure and at constant volume rise in value with temperature. For calculation purposes it is usual to assume an average value of specific heat capacity within the temperature range being considered.

For dry air, the specific heat at constant pressure may be taken as 1.005 kJ/kg-K and the specific heat at constant volume as 0.718 kJ/kg-K. In general the value of γ for dry air is of the order 1.4.