The molar specific heat of oxygen at constant pressure `C_(P) = 7.03 cal//mol .^(@)C` and `R = 8.31 J//mol .^(@)C`. The amount of heat taken by 5 mol of oxygen when heated at constant volume from `10^(@)C` to `20^(@)C` will be approximately.

Options

A. 25 cal
B. 50 cal
C. 250 cal
D. 500 cal

Explanation

To solve the problem, we need to calculate the amount of heat taken by 5 moles of oxygen when heated at constant volume from \(10^\circ C\) to \(20^\circ C\). We will use the specific heat at constant volume (\(C_v\)) and the formula for heat transfer at constant volume.

### Step-by-Step Solution:

1. **Identify Given Values:**
- Molar specific heat at constant pressure, \(C_p = 7.03 \, \text{cal/mol} \cdot ^\circ C\)
- Universal gas constant, \(R = 8.31 \, \text{J/mol} \cdot ^\circ C\)
- Number of moles, \(n = 5 \, \text{mol}\)
- Initial temperature, \(T_1 = 10^\circ C\)
- Final temperature, \(T_2 = 20^\circ C\)

2. **Convert \(R\) from Joules to Calories:**
- We know that \(1 \, \text{cal} = 4.2 \, \text{J}\).
- Therefore, \(R\) in calories is:
\[ R = \frac{8.31 \, \text{J}}{4.2 \, \text{J/cal}} \approx 1.976 \, \text{cal/mol} \cdot ^\circ C \]

3. **Calculate \(C_v\) using Mayer's Relation:**
- Mayer's relation states that:
\[ C_p - C_v = R \]
- Rearranging gives us:
\[ C_v = C_p - R \]
- Substituting the values:
\[ C_v = 7.03 \, \text{cal/mol} \cdot ^\circ C - 1.976 \, \text{cal/mol} \cdot ^\circ C \approx 5.054 \, \text{cal/mol} \cdot ^\circ C \]

4. **Calculate the Change in Temperature (\(\Delta T\)):**
- \(\Delta T = T_2 - T_1 = 20^\circ C - 10^\circ C = 10^\circ C\)

5. **Calculate the Heat (\(Q\)) at Constant Volume:**
- The formula for heat added at constant volume is:
\[ Q = n C_v \Delta T \]
- Substituting the values:
\[ Q = 5 \, \text{mol} \times 5.054 \, \text{cal/mol} \cdot ^\circ C \times 10^\circ C \]
- Calculating \(Q\):
\[ Q = 5 \times 5.054 \times 10 = 252.7 \, \text{cal} \]
- Rounding to the nearest whole number gives:
\[ Q \approx 253 \, \text{cal} \]

### Final Answer:
The amount of heat taken by 5 moles of oxygen when heated at constant volume from \(10^\circ C\) to \(20^\circ C\) is approximately **253 calories**.