Question

The ratio of specific heats $C_{p}/C_{v}$ for a mixture of 1 mole of helium and 1 mole of hydrogen is:

• A. 1.5
• B. 1.67
• C. 1.4
• D. 1.33

Hint:

For mixtures, always find the average $C_{v}$ first, then find $C_{p}$ and divide.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
For a mixture of gases, the effective molar heat capacities are weighted averages of the molar heat capacities of individual gases.
Step 2: Key Formula or Approach:
1. For monatomic gas (He): \(C_{v1} = \frac{3}{2}R\).
2. For diatomic gas (\(\text{H}_2\)): \(C_{v2} = \frac{5}{2}R\).
3. \(C_{v,mix} = \frac{n_1 C_{v1} + n_2 C_{v2}}{n_1 + n_2}\).
4. \(C_{p,mix} = C_{v,mix} + R\).
Step 3: Detailed Explanation:
Given \(n_1 = 1\) (He) and \(n_2 = 1\) (\(\text{H}_2\)).
Calculate \(C_{v,mix}\):
\[ C_{v,mix} = \frac{1 \cdot (1.5R) + 1 \cdot (2.5R)}{1 + 1} = \frac{4R}{2} = 2R \]
Now find \(C_{p,mix}\):
\[ C_{p,mix} = C_{v,mix} + R = 2R + R = 3R \]
Calculate the ratio \(\gamma\):
\[ \gamma = \frac{C_p}{C_v} = \frac{3R}{2R} = 1.5 \]
Step 4: Final Answer:
The ratio of specific heats is 1.5.