If `C_(p) and C_(v)` denote the specific heats (per unit mass of an ideal gas of molecular weight `M`), then
where `R` is the molar gas constant.

Options

A. `C_(p)-C_(v)= R//M^(2)`
B. `C_(p)-C_(v)=R`
C. `C_(p)-C_(v)=RM`
D. `C_(p)-C_(v)= MR`

Explanation

To derive the relationship between the molar specific heats \( C_p \) and \( C_v \) for an ideal gas, we can follow these steps:

### Step-by-Step Solution:

1. **Define Molar Specific Heats**:
The molar specific heats at constant pressure and constant volume are denoted as \( C_p \) and \( C_v \) respectively. These are related to the specific heats per unit mass (denoted as \( c_p \) and \( c_v \)) by the molecular weight \( M \) of the gas.

2. **Relate Molar and Specific Heats**:
The relationship can be expressed as:
\[ C_p = M \cdot c_p \]
\[ C_v = M \cdot c_v \]

3. **Find the Difference**:
We want to find the difference \( C_p - C_v \):
\[ C_p - C_v = M \cdot c_p - M \cdot c_v \]
\[ C_p - C_v = M (c_p - c_v) \]

4. **Use the Known Relation**:
For an ideal gas, it is known that the difference between the molar specific heats is equal to the molar gas constant \( R \):
\[ C_p - C_v = R \]

5. **Combine the Equations**:
From the previous steps, we can equate the two expressions for \( C_p - C_v \):
\[ M (c_p - c_v) = R \]

6. **Solve for \( C_p - C_v \)**:
Rearranging gives:
\[ c_p - c_v = \frac{R}{M} \]
Thus, we have the final relationship:
\[ C_p - C_v = R \]

### Final Result:
The relationship between the molar specific heats \( C_p \) and \( C_v \) for an ideal gas is given by:
\[ C_p - C_v = R \]