A gaseous mixture contains an equal number of hydrogen and nitrogen molecules. Specific heat measurements on this mixture at a temperature below 150 K would indicate the value of the ratio of specific heats for the mixture as

Options

A. `3/2`
B. `4/3`
C. `5/3`
D. `7/5`

Answer

To solve the problem of finding the ratio of specific heats (Cp/Cv) for a gaseous mixture of hydrogen and nitrogen at a temperature below 150 K, we can follow these steps:

### Step 1: Identify the gases and their properties
- We have a mixture of hydrogen (H2) and nitrogen (N2).
- Both hydrogen and nitrogen are diatomic gases.

### Step 2: Determine the degrees of freedom
- For diatomic gases, the degrees of freedom (f) is typically 5 (3 translational + 2 rotational).
- Therefore, the specific heat capacities can be calculated using the formulas:
- \( C_v = \frac{f}{2} R \)
- \( C_p = C_v + R \)

### Step 3: Calculate Cv for both gases
- For hydrogen (H2):
\[ C_{v1} = \frac{5}{2} R \]
- For nitrogen (N2):
\[ C_{v2} = \frac{5}{2} R \]

### Step 4: Calculate Cp for both gases
- For hydrogen (H2):
\[ C_{p1} = C_{v1} + R = \frac{5}{2} R + R = \frac{7}{2} R \]
- For nitrogen (N2):
\[ C_{p2} = C_{v2} + R = \frac{5}{2} R + R = \frac{7}{2} R \]

### Step 5: Calculate the ratio of specific heats for the mixture
- Since the number of molecules of H2 and N2 is equal, we can denote the number of molecules as \( n \).
- The total specific heat capacities for the mixture can be expressed as:
\[ C_p = \frac{n C_{p1} + n C_{p2}}{n + n} = \frac{n \left(\frac{7}{2} R\right) + n \left(\frac{7}{2} R\right)}{2n} = \frac{7R}{2} \]
- Similarly, for Cv:
\[ C_v = \frac{n C_{v1} + n C_{v2}}{n + n} = \frac{n \left(\frac{5}{2} R\right) + n \left(\frac{5}{2} R\right)}{2n} = \frac{5R}{2} \]

### Step 6: Calculate the ratio \( \frac{C_p}{C_v} \)
- Now, we can find the ratio:
\[ \frac{C_p}{C_v} = \frac{\frac{7}{2} R}{\frac{5}{2} R} = \frac{7}{5} \]

### Conclusion
The value of the ratio of specific heats for the mixture of hydrogen and nitrogen at a temperature below 150 K is:
\[ \frac{C_p}{C_v} = \frac{7}{5} \]