Forty calories of heat is needed to raise the temperature of 1 mol of an ideal monatomic gas from `20^(@)C` to `30^(@)C` at a constant pressure. The amount of heat required to raise its temperature over the same interval at a constant volume `(R = 2 cal mol^(-1) K^(-1))` is

Options

A. 20 cal
B. 40 cal
C. 60 cal
D. 80 cal

Explanation

To solve the problem, we need to find the amount of heat required to raise the temperature of 1 mole of an ideal monatomic gas from 20°C to 30°C at constant volume. We will use the information provided in the question and apply the relevant thermodynamic equations.

### Step-by-Step Solution:

1. **Identify the Given Data:**
- Heat at constant pressure (Qp) = 40 calories
- Temperature change (ΔT) = 30°C - 20°C = 10°C = 10 K (since the change in temperature is the same in Celsius and Kelvin)
- R (gas constant) = 2 cal mol^(-1) K^(-1)

2. **Calculate the Heat Capacity at Constant Pressure (Cp):**
- The formula for heat at constant pressure is given by:
\[ Q_p = n C_p \Delta T \]
- Here, \( n = 1 \) mole, and \( \Delta T = 10 \) K. Plugging in the values:
\[ 40 = 1 \cdot C_p \cdot 10 \]
- Rearranging to find \( C_p \):
\[ C_p = \frac{40}{10} = 4 \text{ cal mol}^{-1} \text{ K}^{-1} \]

3. **Use Mayer's Relation to Find Cv:**
- Mayer's relation states:
\[ C_p - C_v = R \]
- We already found \( C_p = 4 \) cal mol^(-1) K^(-1) and \( R = 2 \) cal mol^(-1) K^(-1). Now, substituting these values:
\[ C_v = C_p - R = 4 - 2 = 2 \text{ cal mol}^{-1} \text{ K}^{-1} \]

4. **Calculate the Heat at Constant Volume (Qv):**
- The formula for heat at constant volume is given by:
\[ Q_v = n C_v \Delta T \]
- Substituting \( n = 1 \) mole, \( C_v = 2 \) cal mol^(-1) K^(-1), \) and \( \Delta T = 10 \) K:
\[ Q_v = 1 \cdot 2 \cdot 10 = 20 \text{ calories} \]

5. **Conclusion:**
- The amount of heat required to raise the temperature of the gas at constant volume is **20 calories**.