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 find the ratio of specific heats (Cp/Cv) for a gaseous mixture of hydrogen and nitrogen, we can follow these steps:

### Step 1: Identify the specific heats of the individual gases
- Hydrogen (H₂) is a diatomic gas with specific heat capacities:
- Cp(H₂) = 5R/2
- Cv(H₂) = 3R/2

- Nitrogen (N₂) is also a diatomic gas with specific heat capacities:
- Cp(N₂) = 7R/2
- Cv(N₂) = 5R/2

### Step 2: Calculate the specific heats for the mixture
Since the mixture contains equal numbers of hydrogen and nitrogen molecules, we can calculate the average specific heats for the mixture.

- The total number of moles (n) of the mixture can be expressed as:
- n = n(H₂) + n(N₂)

Since there are equal numbers of molecules, we can assume:
- n(H₂) = n(N₂) = n₀ (where n₀ is the number of moles of each gas).

Thus, the total number of moles:
- n = n₀ + n₀ = 2n₀

### Step 3: Calculate the average Cp and Cv for the mixture
Using the formula for average specific heats:
- Cp(mixture) = (Cp(H₂) * n(H₂) + Cp(N₂) * n(N₂)) / n
- Cv(mixture) = (Cv(H₂) * n(H₂) + Cv(N₂) * n(N₂)) / n

Substituting the values:
- Cp(mixture) = [(5R/2) * n₀ + (7R/2) * n₀] / (2n₀)
- Cp(mixture) = [(5R/2 + 7R/2) * n₀] / (2n₀)
- Cp(mixture) = (12R/2) / 2 = 3R

- Cv(mixture) = [(3R/2) * n₀ + (5R/2) * n₀] / (2n₀)
- Cv(mixture) = [(3R/2 + 5R/2) * n₀] / (2n₀)
- Cv(mixture) = (8R/2) / 2 = 2R

### Step 4: Calculate the ratio Cp/Cv for the mixture
Now we can find the ratio of specific heats:
- Cp/Cv = (3R) / (2R) = 3/2

### Final Answer
The value of the ratio of specific heats for the mixture (Cp/Cv) is **3/2**.

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