Question

A gaseous mixture consists of \(4\ \text{g}\) of oxygen and \(4\ \text{g}\) of helium. The ratio \(\dfrac{C_P}{C_V}\) of the mixture is
\[ \text{(}C_P \text{ and } C_V \text{ are molar specific heats of the mixture at constant pressure and constant volume respectively.)} \]

• A. \(\dfrac{29}{13}\)
• B. \(\dfrac{47}{18}\)
• C. \(\dfrac{47}{29}\)
• D. \(\dfrac{18}{13}\)

Hint:

For an ideal gas mixture, \[ C_P=\sum n_i C_{P,i}, \qquad C_V=\sum n_i C_{V,i}. \] Always calculate the number of moles first and then use weighted heat capacities of the constituent gases.

Answer 1

The Correct Option is C

Step 1: Find moles of each gas.
\( n_{O_2} = 4/32 = 1/8\,\text{mol} \) (diatomic, \( C_V = 5R/2 \)). \( n_{He} = 4/4 = 1\,\text{mol} \) (monatomic, \( C_V = 3R/2 \)). Total moles \( = 9/8 \).

Step 2: Compute mixture \( C_P/C_V \).
\( (C_V)_{mix} = \frac{(1/8)(5R/2)+(1)(3R/2)}{9/8} = \frac{29R/16}{9/8} = \frac{29R}{18} \). \( (C_P)_{mix} = (C_V)_{mix}+R = \frac{47R}{18} \). \[ \frac{C_P}{C_V} = \frac{47}{29} \] \[ \boxed{\dfrac{47}{29}} \]