The specific heat of a monoatomic gas at constant pressure is 248.2 J `kg^(-1)K^(-1)` and at constant volume it is 149.0 J `kg^(-1) K^(-1)`. Find the mean molar mass of the gas.

Explanation

To find the mean molar mass of the monoatomic gas given its specific heats at constant pressure (Cp) and constant volume (Cv), we can use the relationship between these quantities and the ideal gas constant (R).

### Step-by-Step Solution:

1. **Identify the given values:**
- Specific heat at constant pressure, \( C_p = 248.2 \, \text{J kg}^{-1} \text{K}^{-1} \)
- Specific heat at constant volume, \( C_v = 149.0 \, \text{J kg}^{-1} \text{K}^{-1} \)

2. **Use the relationship between Cp, Cv, and R:**
The relationship for ideal gases is given by:
\[ C_p - C_v = R \]
where \( R \) is the ideal gas constant.

3. **Calculate R:**
\[ R = C_p - C_v = 248.2 \, \text{J kg}^{-1} \text{K}^{-1} - 149.0 \, \text{J kg}^{-1} \text{K}^{-1} \]
\[ R = 99.2 \, \text{J kg}^{-1} \text{K}^{-1} \]

4. **Relate R to molar mass (M):**
The molar heat capacities can be expressed in terms of molar mass (M):
\[ C_p = \frac{R}{M} + C_v \]
Rearranging gives:
\[ M = \frac{R}{C_p - C_v} \]

5. **Substitute the values:**
Now substituting the values we have:
\[ M = \frac{R}{C_p - C_v} = \frac{99.2 \, \text{J kg}^{-1} \text{K}^{-1}}{C_p - C_v} \]
We already calculated \( C_p - C_v = 99.2 \, \text{J kg}^{-1} \text{K}^{-1} \).

6. **Calculate M:**
\[ M = \frac{R}{99.2} = \frac{8.314 \, \text{J mol}^{-1} \text{K}^{-1}}{99.2 \, \text{J kg}^{-1} \text{K}^{-1}} \]
\[ M = 0.0838 \, \text{kg mol}^{-1} \]

### Final Answer:
The mean molar mass of the gas is \( 0.0838 \, \text{kg mol}^{-1} \).

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