Question

If one mole of a monoatomic gas (γ = 5 / 3) is mixed with one mole of a diatomic gas (γ = 7 / 5) the value of γ for the mixture is
1. 1.40
2. 1.50
3. 1.53
4. 3.07

Answer 1

Correct Answer - Option 2 : 1.50

CONCEPT:

• Specific heat capacity at constant pressure (C_P): It is the amount of heat required to raise the temperature of 1 kg of gas maintained at constant pressure by 1 degree Celcius.
• Specific heat capacity at constant volume (CV): It is the amount of heat required to raise the temperature of 1 kg of gas maintained at constant volume by 1 degree Celcius.
• The SI unit for both C_P and C_V is J/(Kg.K).
• \(\gamma\) for a gas is the ratio of the specific heat capacity at constant pressure (C_p) to the specific heat capacity at constant volume (C_V). 

\(\Rightarrow \gamma = \frac{C_P}{C_V}\)

CALCULATION:

• For monoatomic gas,

 \(\gamma^m = \frac{C_P^m}{C_V^m}=\frac{5}{3} \\ C_P^m = \frac{5}{2} RT \\ C_V^m = \frac{3}{2} RT\)

In the above equations, the superscript m is just included to denote monoatomic gas. 

• For diatomic gas,

\(\gamma^d = \frac{C_P^d}{C_V^d}=\frac{7}{5} \\ C_P^d = \frac{7}{2} RT \\ C_V^d = \frac{5}{2} RT\)

In the above equations, the superscript d is just included to denote diatomic gas. 

• For a mixture of one mole of monoatomic gas and one mole of diatomic gas, that is a total of two moles of gas:

\(\Rightarrow C_P (net) = \frac{C_P^m +C_P^d}{1+1} = \frac{\frac{5}{2} RT + \frac{7}{2}RT}{2} = 3RT \\ \Rightarrow C_V (net) = \frac{C_V^m+C_V^d}{1+1}= \frac{\frac{3}{2} RT + \frac{5}{2}RT}{2} = 2RT \\ \Rightarrow \gamma (net) = \frac{C_P(net)}{C_V(net)} = \frac{3RT}{2RT} = 1.5\)

• Therefore, option 2 is correct.