A monatomic ideal gas is heated at constant volume until its pressure is doubled. It is again heated at constant pressure, until its volume is doubled. Find molar specific heat for the whole process.

Options

A. `(3R)/(2)`
B. `(5R)/(2)`
C. `(13R)/(6)`
D. `(7R)/(2)`

Answer

To find the molar specific heat for the entire process described, we will break the process into two parts and calculate the heat added in each part.

### Step 1: Understand the Process
1. **First Part**: The gas is heated at constant volume until its pressure is doubled.
2. **Second Part**: The gas is then heated at constant pressure until its volume is doubled.

### Step 2: Analyze the First Part
- **Initial State**: Let the initial pressure be \( P_1 \), initial volume \( V \), and initial temperature \( T_1 \).
- **Final State**: The pressure is doubled, so \( P_2 = 2P_1 \).
- Using the ideal gas law, \( PV = nRT \), we can express the initial and final states:
\[ P_1 V = n R T_1 \quad \text{(initial state)} \]
\[ P_2 V = n R T_2 \quad \text{(final state)} \]
Substituting \( P_2 \):
\[ 2P_1 V = n R T_2 \]
- From the first equation, we can express \( T_1 \):
\[ T_1 = \frac{P_1 V}{n R} \]
- Now substituting \( T_1 \) into the equation for \( T_2 \):
\[ 2 \left( \frac{P_1 V}{n R} \right) = T_2 \implies T_2 = 2T_1 \]

### Step 3: Calculate Heat Added in First Part
- The heat added at constant volume is given by:
\[ Q_1 = n C_V (T_2 - T_1) \]
- For a monatomic ideal gas, \( C_V = \frac{3}{2} R \):
\[ Q_1 = n \left( \frac{3}{2} R \right) (2T_1 - T_1) = n \left( \frac{3}{2} R \right) T_1 \]

### Step 4: Analyze the Second Part
- **Initial State**: The gas is now at \( T_2 = 2T_1 \) and pressure \( P_2 = 2P_1 \).
- **Final State**: The volume is doubled, so \( V_2 = 2V \).
- Using the ideal gas law again:
\[ P_2 V_2 = n R T_3 \]
Substituting \( P_2 \) and \( V_2 \):
\[ 2P_1 (2V) = n R T_3 \implies 4P_1 V = n R T_3 \]
- From the initial state:
\[ P_1 V = n R T_1 \implies T_3 = 4T_1 \]

### Step 5: Calculate Heat Added in Second Part
- The heat added at constant pressure is given by:
\[ Q_2 = n C_P (T_3 - T_2) \]
- For a monatomic ideal gas, \( C_P = \frac{5}{2} R \):
\[ Q_2 = n \left( \frac{5}{2} R \right) (4T_1 - 2T_1) = n \left( \frac{5}{2} R \right) (2T_1) = n \left( 5 R \right) T_1 \]

### Step 6: Total Heat Added
- The total heat added \( Q \) is:
\[ Q = Q_1 + Q_2 = n \left( \frac{3}{2} R T_1 \right) + n \left( 5 R T_1 \right) = n \left( \frac{3}{2} R + 5 R \right) T_1 = n \left( \frac{13}{2} R \right) T_1 \]

### Step 7: Calculate Total Change in Temperature
- The total change in temperature from \( T_1 \) to \( T_3 \):
\[ \Delta T = T_3 - T_1 = 4T_1 - T_1 = 3T_1 \]

### Step 8: Molar Specific Heat for the Whole Process
- The molar specific heat \( C \) for the whole process is defined as:
\[ C = \frac{Q}{n \Delta T} = \frac{n \left( \frac{13}{2} R \right) T_1}{n (3T_1)} = \frac{13}{6} R \]

### Final Answer
The molar specific heat for the whole process is:
\[ C = \frac{13}{6} R \]