An ideal gas having molar specific heat capaicty at constatnt volume is `3/2`R, the molar specific heat capacities at constant pressure is

Options

A. `(1)/(2)R`
B. `(5)/(2)R`
C. `(7)/(2)R`
D. `(9)/(2)R`

Explanation

To find the molar specific heat capacity at constant pressure (Cp) for an ideal gas when the molar specific heat capacity at constant volume (Cv) is given as \( \frac{3}{2} R \), we can use the relationship known as Mayer's formula:

### Step 1: Write down Mayer's formula
Mayer's formula states that:
\[ C_p - C_v = R \]

### Step 2: Substitute the given value of Cv
We know that \( C_v = \frac{3}{2} R \). Substituting this into Mayer's formula gives:
\[ C_p - \frac{3}{2} R = R \]

### Step 3: Solve for Cp
Now, we can solve for \( C_p \) by adding \( \frac{3}{2} R \) to both sides:
\[ C_p = R + \frac{3}{2} R \]

### Step 4: Combine the terms
To combine the terms on the right side, we can express \( R \) as \( \frac{2}{2} R \):
\[ C_p = \frac{2}{2} R + \frac{3}{2} R = \frac{5}{2} R \]

### Conclusion
Thus, the molar specific heat capacity at constant pressure \( C_p \) is:
\[ C_p = \frac{5}{2} R \]

### Final Answer
The molar specific heat capacity at constant pressure is \( \frac{5}{2} R \).

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