One mole of monoatomic gas is mixed with one mole of a diatomic ideal gas. The molar specific heat of the mixture at constant volume is ___________________.

Answer

To find the molar specific heat of the mixture at constant volume for one mole of a monoatomic gas and one mole of a diatomic ideal gas, we can follow these steps:

### Step 1: Identify the specific heats of the gases
- The molar specific heat at constant volume (C_v) for a monoatomic ideal gas is given by:
\[ C_{v1} = \frac{3}{2} R \]
- The molar specific heat at constant volume (C_v) for a diatomic ideal gas is given by:
\[ C_{v2} = \frac{5}{2} R \]

### Step 2: Determine the number of moles
- We have:
- \( n_1 = 1 \) mole of monoatomic gas
- \( n_2 = 1 \) mole of diatomic gas

### Step 3: Use the formula for the molar specific heat of the mixture
The formula for the molar specific heat at constant volume for a mixture of ideal gases is:
\[ C_{v,\text{mixture}} = \frac{n_1 C_{v1} + n_2 C_{v2}}{n_1 + n_2} \]

### Step 4: Substitute the values into the formula
Substituting the values we have:
\[ C_{v,\text{mixture}} = \frac{1 \cdot \frac{3}{2} R + 1 \cdot \frac{5}{2} R}{1 + 1} \]

### Step 5: Simplify the expression
Calculating the numerator:
\[ C_{v,\text{mixture}} = \frac{\frac{3}{2} R + \frac{5}{2} R}{2} = \frac{\frac{8}{2} R}{2} = \frac{4R}{2} = 2R \]

### Final Answer
Thus, the molar specific heat of the mixture at constant volume is:
\[ \boxed{2R} \]