To solve the problem, we need to analyze the relationship between the specific heats at constant pressure (C_P) and constant volume (C_V) for hydrogen and nitrogen.
### Step 1: Understand the relationship
We know that for any gas, the difference between the specific heats at constant pressure and constant volume is given by:
\[ C_P - C_V = R \]
where \( R \) is the gas constant. However, in this problem, we are given that:
- For hydrogen, \( C_P - C_V = m \)
- For nitrogen, \( C_P - C_V = n \)
### Step 2: Use the molecular weights
The molecular weight of hydrogen (H₂) is given as 2, and for nitrogen (N₂), it is given as 14. The specific heat capacities can be expressed in terms of the molecular weights:
- For hydrogen, we can denote the specific heat capacities as \( C_{P,H} \) and \( C_{V,H} \).
- For nitrogen, we denote them as \( C_{P,N} \) and \( C_{V,N} \).
### Step 3: Relate m and n
From the problem, we have:
\[ m = C_{P,H} - C_{V,H} \]
\[ n = C_{P,N} - C_{V,N} \]
### Step 4: Express the specific heats using molecular weights
The specific heats at constant volume for ideal gases can be related to their molecular weights:
\[ C_V \propto \frac{1}{M} \]
where \( M \) is the molecular weight. Therefore, we can express:
- For hydrogen:
\[ C_{V,H} = k \cdot \frac{1}{2} \]
- For nitrogen:
\[ C_{V,N} = k \cdot \frac{1}{14} \]
### Step 5: Calculate the difference for both gases
Using the relationship \( C_P - C_V = R \) and substituting for \( C_V \):
- For hydrogen:
\[ m = C_{P,H} - C_{V,H} = C_{P,H} - k \cdot \frac{1}{2} \]
- For nitrogen:
\[ n = C_{P,N} - C_{V,N} = C_{P,N} - k \cdot \frac{1}{14} \]
### Step 6: Find the ratio of m to n
To find the relationship between \( m \) and \( n \):
\[ \frac{m}{n} = \frac{C_{P,H} - k \cdot \frac{1}{2}}{C_{P,N} - k \cdot \frac{1}{14}} \]
### Step 7: Simplify the ratio
Assuming that the specific heats at constant pressure are proportional to the molecular weights, we can simplify:
\[ \frac{m}{n} = \frac{14}{2} = 7 \]
### Conclusion
Thus, the relationship between \( m \) and \( n \) is:
\[ m = 7n \]
