One mole of an ideal monoatomic gas is initially at 300K. Find the final temperature if 200J of heat are added as follows:
(a) at constant volume (b) at constant pressure.

Explanation

To solve the problem, we need to find the final temperature of one mole of an ideal monoatomic gas after adding 200 J of heat under two different conditions: at constant volume and at constant pressure.

### (a) At Constant Volume

1. **Identify the formula for heat added at constant volume**:
The heat added at constant volume is given by the equation:
\[ Q = n C_V \Delta T \]
where:
- \( Q \) is the heat added (200 J),
- \( n \) is the number of moles (1 mole),
- \( C_V \) is the molar heat capacity at constant volume for a monoatomic gas, which is \( \frac{3}{2} R \),
- \( \Delta T \) is the change in temperature.

2. **Substitute known values**:
The value of the gas constant \( R \) is approximately \( 8.31 \, \text{J/(mol K)} \). Thus, we can calculate \( C_V \):
\[ C_V = \frac{3}{2} R = \frac{3}{2} \times 8.31 = 12.465 \, \text{J/(mol K)} \]

3. **Set up the equation**:
Now we can substitute the values into the heat equation:
\[ 200 = 1 \times 12.465 \times (T_f - 300) \]

4. **Solve for \( T_f \)**:
Rearranging the equation gives:
\[ T_f - 300 = \frac{200}{12.465} \]
\[ T_f - 300 = 16.04 \]
\[ T_f = 316.04 \, \text{K} \]

### (b) At Constant Pressure

1. **Identify the formula for heat added at constant pressure**:
The heat added at constant pressure is given by:
\[ Q = n C_P \Delta T \]
where:
- \( C_P \) is the molar heat capacity at constant pressure for a monoatomic gas, which is \( \frac{5}{2} R \).

2. **Calculate \( C_P \)**:
\[ C_P = \frac{5}{2} R = \frac{5}{2} \times 8.31 = 20.825 \, \text{J/(mol K)} \]

3. **Set up the equation**:
Substitute the values into the heat equation:
\[ 200 = 1 \times 20.825 \times (T_f - 300) \]

4. **Solve for \( T_f \)**:
Rearranging gives:
\[ T_f - 300 = \frac{200}{20.825} \]
\[ T_f - 300 = 9.61 \]
\[ T_f = 309.61 \, \text{K} \]

### Final Results
- (a) Final temperature at constant volume: **316.04 K**
- (b) Final temperature at constant pressure: **309.61 K**