For an ideal gas `(C_(p,m))/(C_(v,m))=gamma`. The molecular mass of the gas is M, its specific heat capacity at constant volume is :

Options

A. `(R)/(M(gamma-1))`
B. `(M)/(R(gamma-1))`
C. `(gammaRM)/(gamma-1)`
D. `(gammaR)/(M(gamma-1)`

Explanation

To find the specific heat capacity at constant volume (C) for an ideal gas, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the relationship between Cp and Cv**:
We know that for an ideal gas, the ratio of specific heat capacities at constant pressure (Cp) and constant volume (Cv) is given by:
\[ \frac{C_{p,m}}{C_{v,m}} = \gamma \]
where \( \gamma \) is the heat capacity ratio.

2. **Using the relationship between Cp and Cv**:
There is a fundamental relationship for ideal gases:
\[ C_p - C_v = R \]
where R is the universal gas constant.

3. **Express Cp in terms of Cv**:
Rearranging the above equation gives:
\[ C_p = C_v + R \]
Let's label this as Equation (1).

4. **Substituting Cp in the ratio**:
Substitute Equation (1) into the ratio given in the problem:
\[ \frac{C_v + R}{C_v} = \gamma \]
This simplifies to:
\[ \frac{R}{C_v} + 1 = \gamma \]

5. **Isolating R/Cv**:
Rearranging gives:
\[ \frac{R}{C_v} = \gamma - 1 \]

6. **Finding Cv**:
From the above equation, we can express Cv as:
\[ C_v = \frac{R}{\gamma - 1} \]
This is the molar heat capacity at constant volume.

7. **Relating Cv to specific heat capacity (C)**:
The molar heat capacity (Cv) can also be expressed in terms of the specific heat capacity (C) and the molecular mass (M) of the gas:
\[ C_v = M \cdot C \]
Thus, we have:
\[ M \cdot C = \frac{R}{\gamma - 1} \]

8. **Solving for C**:
Finally, we can solve for the specific heat capacity (C):
\[ C = \frac{R}{M(\gamma - 1)} \]

### Final Answer:
The specific heat capacity at constant volume is:
\[ C = \frac{R}{M(\gamma - 1)} \]