Heat Capacity Ratio

To find C_P/C_V of a gas, one sometimes uses the following method. A certain amount of gas with initial temperature T₀, pressure P₀, and volume V₀, is heated by a current flowing through a platinum wire for a time t. The experiment is done twice: first at a constant volume V₀ with the pressure changing from P₀ to P₁, and then at a constant pressure with the volume changing from V₀ to V₁ The time t is the same in both experiments. Find the ratio C_P/C_V (the gas may be considered ideal).

SOLUTION

If the gas is heated at constant volume, then the amount of heat Q transferred to the gas is

(1)

where c_v is the heat capacity by weight of the gas, m is the mass, and T is the temperature at pressure P₁. Using the ideal gas law at the beginning and end of heating gives

(2)

where n is the number of moles of the gas. From (1) and (2),

(3)

and

(4)

For heating at constant pressure,

(5)

Similarly,

(6)

So

(7)

and

(8)

Since the time t during which the current flows through the wire is the same in both experiments, the amount of heat transferred to the gas is also the same: Q' = Q. Equating (4) and (8), we obtain

(9)