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 step by step, we will use the concepts of thermodynamics related to heat transfer in ideal gases.

### Step 1: Identify the given values
- Heat required at constant pressure (Q_p) = 50 cal
- Number of moles (n) = 1 mole
- Change in temperature (ΔT) = 25°C - 20°C = 5°C = 5 K
- Gas constant (R) = 2 cal/mol/K

### Step 2: Use the formula for heat transfer at constant pressure
The formula for heat transfer at constant pressure is given by:
\[ Q_p = n C_p \Delta T \]
Substituting the known values:
\[ 50 \, \text{cal} = 1 \, \text{mol} \times C_p \times 5 \, \text{K} \]

### Step 3: Solve for \( C_p \)
Rearranging the equation to find \( C_p \):
\[ C_p = \frac{50 \, \text{cal}}{1 \, \text{mol} \times 5 \, \text{K}} \]
\[ C_p = 10 \, \text{cal/mol/K} \]

### Step 4: Use the relation between \( C_p \) and \( C_v \)
The relation between specific heats at constant pressure and constant volume is given by:
\[ C_p - C_v = R \]
Substituting the known values:
\[ 10 \, \text{cal/mol/K} - C_v = 2 \, \text{cal/mol/K} \]
Rearranging to find \( C_v \):
\[ C_v = 10 \, \text{cal/mol/K} - 2 \, \text{cal/mol/K} \]
\[ C_v = 8 \, \text{cal/mol/K} \]

### Step 5: Calculate the heat transfer at constant volume
Now we will calculate the heat transfer at constant volume using the formula:
\[ Q_v = n C_v \Delta T \]
Substituting the values:
\[ Q_v = 1 \, \text{mol} \times 8 \, \text{cal/mol/K} \times 5 \, \text{K} \]
\[ Q_v = 40 \, \text{cal} \]

### Conclusion
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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