Temperature of 4 moles of gas increases from 300 K to 500 K find `C_v` if `DeltaU = 5000 J`.

Options

A. (a) 2.13
B. (b) 4.75
C. (c) 6.25
D. (d) 8.25

Explanation

To solve the problem, we need to find the specific heat capacity at constant volume, \( C_v \), for 4 moles of gas when the temperature changes from 300 K to 500 K and the change in internal energy, \( \Delta U \), is given as 5000 J.

### Step-by-Step Solution:

1. **Identify the given values:**
- Number of moles, \( n = 4 \)
- Initial temperature, \( T_1 = 300 \, \text{K} \)
- Final temperature, \( T_2 = 500 \, \text{K} \)
- Change in internal energy, \( \Delta U = 5000 \, \text{J} \)

2. **Calculate the change in temperature (\( \Delta T \)):**
\[ \Delta T = T_2 - T_1 = 500 \, \text{K} - 300 \, \text{K} = 200 \, \text{K} \]

3. **Use the formula for change in internal energy:**
The relationship between change in internal energy, number of moles, specific heat at constant volume, and change in temperature is given by:
\[ \Delta U = n C_v \Delta T \]
Rearranging this formula to solve for \( C_v \):
\[ C_v = \frac{\Delta U}{n \Delta T} \]

4. **Substitute the known values into the equation:**
\[ C_v = \frac{5000 \, \text{J}}{4 \, \text{moles} \times 200 \, \text{K}} \]

5. **Calculate \( C_v \):**
\[ C_v = \frac{5000}{800} = 6.25 \, \text{J/(K·mol)} \]

### Final Answer:
The specific heat capacity at constant volume, \( C_v \), is:
\[ C_v = 6.25 \, \text{J/(K·mol)} \]