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

Explanation

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

### Step-by-Step Solution:

1. **Understand the Relationship**:
We know that the ratio of specific heats is defined as:
\[ \gamma = \frac{C_p}{C_v} \]
This is our first equation.

2. **Use the Relation Between \( C_p \) and \( C_v \)**:
There is a known relation between \( C_p \) and \( C_v \):
\[ C_p - C_v = R \]
where \( R \) is the universal gas constant.

3. **Express \( C_p \) in Terms of \( C_v \)**:
From the equation \( C_p - C_v = R \), we can express \( C_p \) as:
\[ C_p = C_v + R \]

4. **Substitute \( C_p \) in the Ratio Equation**:
Substitute \( C_p \) in the first equation:
\[ \gamma = \frac{C_v + R}{C_v} \]
This can be rewritten as:
\[ \gamma = 1 + \frac{R}{C_v} \]

5. **Rearranging to Find \( C_v \)**:
Rearranging the equation to isolate \( C_v \):
\[ \gamma - 1 = \frac{R}{C_v} \]
Therefore, we can express \( C_v \) as:
\[ C_v = \frac{R}{\gamma - 1} \]

6. **Find \( C_p \)**:
Now substitute \( C_v \) back into the equation for \( C_p \):
\[ C_p = C_v + R = \frac{R}{\gamma - 1} + R \]
To combine these terms, we can express \( R \) with a common denominator:
\[ C_p = \frac{R}{\gamma - 1} + \frac{R(\gamma - 1)}{\gamma - 1} = \frac{R + R(\gamma - 1)}{\gamma - 1} \]
Simplifying gives:
\[ C_p = \frac{\gamma R}{\gamma - 1} \]

7. **Calculate the Specific Heat Capacity**:
The specific heat capacity at constant pressure per unit mass is given by:
\[ \text{Specific Heat Capacity} = \frac{C_p}{M} \]
Substituting \( C_p \):
\[ \text{Specific Heat Capacity} = \frac{\frac{\gamma R}{\gamma - 1}}{M} = \frac{\gamma R}{M(\gamma - 1)} \]

### Final Result:
Thus, the specific heat capacity at constant pressure is:
\[ C_p = \frac{\gamma R}{M(\gamma - 1)} \]