50 cal of heat is required to raise the temperature of 1 mole of an ideal gas from `20^@ C` to `25^@ C`, while the pressure of the gas is kept constant. The amount of heat required to raise the temperature of the same has through same temperature range at constant volume is ( R = 2 cal/mol/K)

Options

A. 70 cal
B. 60 cal
C. 40 cal
D. 50 cal

Answer

To solve the problem, we need to find the amount of heat required to raise the temperature of 1 mole of an ideal gas from 20°C to 25°C at constant volume (isochoric process). We know that the heat required at constant pressure (isobaric process) is given as 50 cal.

### Step-by-Step Solution:

1. **Identify the given data:**
- Heat required at constant pressure, \( Q_P = 50 \, \text{cal} \)
- Number of moles, \( n = 1 \, \text{mol} \)
- Temperature change, \( \Delta T = 25°C - 20°C = 5°C \) (which is equivalent to 5 K)

2. **Use the formula for heat at constant pressure:**
\[ Q_P = n C_P \Delta T \]
Where \( C_P \) is the molar heat capacity at constant pressure.

3. **Substituting the known values into the equation:**
\[ 50 = 1 \cdot C_P \cdot 5 \]
This simplifies to:
\[ C_P = \frac{50}{5} = 10 \, \text{cal/mol/K} \]

4. **Relate \( C_P \) to \( C_V \):**
The relationship between \( C_P \) and \( C_V \) for an ideal gas is given by:
\[ C_P = C_V + R \]
Where \( R \) is the gas constant. Given \( R = 2 \, \text{cal/mol/K} \), we can find \( C_V \):
\[ C_V = C_P - R = 10 - 2 = 8 \, \text{cal/mol/K} \]

5. **Calculate the heat required at constant volume:**
Using the formula for heat at constant volume:
\[ Q_V = n C_V \Delta T \]
Substituting the known values:
\[ Q_V = 1 \cdot 8 \cdot 5 = 40 \, \text{cal} \]

### Final Answer:
The amount of heat required to raise the temperature of the gas from 20°C to 25°C at constant volume is **40 cal**.

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