Calculate the value of mechanical equivalent of heat from the following data. Specific heat capacity of air at constant volume and at constant pressure are `4.93 cal//mol-K` and `6.90 cal//mol-K` respectively. Gas constant `R = 8.3 J//mol-K`.

Explanation

To calculate the mechanical equivalent of heat (denoted as \( J \)), we can use the relationship between the specific heat capacities at constant pressure (\( C_p \)) and constant volume (\( C_v \)), along with the gas constant \( R \).

### Step-by-Step Solution

1. **Identify the given values:**
- Specific heat capacity at constant volume, \( C_v = 4.93 \, \text{cal/mol-K} \)
- Specific heat capacity at constant pressure, \( C_p = 6.90 \, \text{cal/mol-K} \)
- Gas constant, \( R = 8.3 \, \text{J/mol-K} \)

2. **Use the relationship between \( C_p \), \( C_v \), and \( R \):**
\[ C_p - C_v = R \]
Substitute the values of \( C_p \) and \( C_v \):
\[ 6.90 \, \text{cal/mol-K} - 4.93 \, \text{cal/mol-K} = R \]
\[ 1.97 \, \text{cal/mol-K} = R \]

3. **Convert \( R \) from Joules to calories:**
We know that \( R = 8.3 \, \text{J/mol-K} \) and we need to express \( 1.97 \, \text{cal/mol-K} \) in Joules. The conversion factor is:
\[ 1 \, \text{cal} = 4.184 \, \text{J} \]
Therefore, convert \( 1.97 \, \text{cal/mol-K} \) to Joules:
\[ 1.97 \, \text{cal/mol-K} \times 4.184 \, \text{J/cal} = 8.25 \, \text{J/mol-K} \]

4. **Calculate the mechanical equivalent of heat \( J \):**
The mechanical equivalent \( J \) is given by the ratio of \( R \) in Joules to \( C_v \) in calories:
\[ J = \frac{R}{C_v} \]
Substitute the values:
\[ J = \frac{8.3 \, \text{J/mol-K}}{1.97 \, \text{cal/mol-K}} \]

5. **Perform the calculation:**
\[ J \approx \frac{8.3}{1.97} \approx 4.21 \, \text{J/cal} \]

### Final Answer
The mechanical equivalent of heat \( J \) is approximately \( 4.21 \, \text{J/cal} \).