If for hydrogen `C_(P) - C_(V) = m` and for nitrogen `C_(P) - C_(V) = n`, where `C_(P)` and `C_(V)` refer to specific heats per unit mass respectively at constant pressure and constant volume, the relation between `m` and `n` is (molecular weight of hydrogen = 2 and molecular weight or nitrogen = 14)

Options

A. `n = 14 m`
B. `n = 7m`
C. `m = 7n`
D. `m = 14 n`

Explanation

To solve the problem, we need to find the relationship between \( m \) and \( n \) given the specific heat capacities of hydrogen and nitrogen.

### Step-by-Step Solution:

1. **Understanding the Definitions**:
- The specific heat at constant pressure is denoted as \( C_P \).
- The specific heat at constant volume is denoted as \( C_V \).
- The difference \( C_P - C_V \) is related to the heat added to the system and the temperature change.

2. **Given Information**:
- For hydrogen: \( C_P - C_V = m \)
- For nitrogen: \( C_P - C_V = n \)
- Molecular weight of hydrogen (\( M_H \)) = 2
- Molecular weight of nitrogen (\( M_N \)) = 14

3. **Expressing the Heat Transfer**:
- For hydrogen, we can express the heat transfer as:
\[ C_P - C_V = \frac{\Delta Q}{M_H \cdot \Delta T} = m \]
- For nitrogen, we can express the heat transfer as:
\[ C_P - C_V = \frac{\Delta Q}{M_N \cdot \Delta T} = n \]

4. **Setting Up the Equations**:
- From the equation for hydrogen:
\[ \Delta Q = m \cdot M_H \cdot \Delta T = m \cdot 2 \cdot \Delta T \]
- From the equation for nitrogen:
\[ \Delta Q = n \cdot M_N \cdot \Delta T = n \cdot 14 \cdot \Delta T \]

5. **Equating the Two Expressions for Heat Transfer**:
- Since both expressions equal \( \Delta Q \), we can set them equal to each other:
\[ m \cdot 2 \cdot \Delta T = n \cdot 14 \cdot \Delta T \]

6. **Cancelling \( \Delta T \)**:
- Assuming \( \Delta T \) is not zero, we can divide both sides by \( \Delta T \):
\[ m \cdot 2 = n \cdot 14 \]

7. **Solving for the Relationship Between \( m \) and \( n \)**:
- Rearranging the equation gives:
\[ m = \frac{14}{2} n = 7n \]

8. **Conclusion**:
- The relationship between \( m \) and \( n \) is:
\[ m = 7n \]

### Final Answer:
The relation between \( m \) and \( n \) is \( m = 7n \).