The ratio `(C_p)/(C_v)=gamma` for a gas. Its molecular weight is M. Its specific heat capacity at constant pressure is

Options

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

Explanation

To find the specific heat capacity at constant pressure (\(C_p\)) for a gas given the ratio \(\frac{C_p}{C_v} = \gamma\) and its molecular weight \(M\), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between \(C_p\) and \(C_v\)**:
The relationship between the specific heats at constant pressure and constant volume is given by:
\[ C_p - C_v = R \]
where \(R\) is the gas constant.

2. **Express \(C_v\) in terms of \(\gamma\)**:
From the ratio \(\frac{C_p}{C_v} = \gamma\), we can express \(C_p\) as:
\[ C_p = \gamma C_v \]
Substituting this into the first equation gives:
\[ \gamma C_v - C_v = R \]
Simplifying this, we get:
\[ C_v(\gamma - 1) = R \]
Thus, we can express \(C_v\) as:
\[ C_v = \frac{R}{\gamma - 1} \]

3. **Substitute \(C_v\) back to find \(C_p\)**:
Now substituting \(C_v\) back into the expression for \(C_p\):
\[ C_p = \gamma C_v = \gamma \left(\frac{R}{\gamma - 1}\right) \]
This simplifies to:
\[ C_p = \frac{\gamma R}{\gamma - 1} \]

4. **Calculate specific heat capacity at constant pressure**:
The specific heat capacity at constant pressure per unit mass (\(S_p\)) is given by:
\[ S_p = \frac{C_p}{M} \]
Substituting the expression for \(C_p\):
\[ S_p = \frac{\frac{\gamma R}{\gamma - 1}}{M} \]
This simplifies to:
\[ S_p = \frac{\gamma R}{M(\gamma - 1)} \]

### Final Answer:
The specific heat capacity at constant pressure per unit mass is:
\[ S_p = \frac{\gamma R}{M(\gamma - 1)} \]