At constant volume , 4 moles of an ideal gas when heated from 300 K to 500 K change its internal energy by 5000 J. The molar heat capacity at constant volume is ?

Options

A. `3.125`
B. `6.25`
C. `6.75`
D. `-6.75`

Answer

To find the molar heat capacity at constant volume (\(C_v\)) for the given ideal gas, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Number of moles (\(n\)) = 4 moles
- Initial temperature (\(T_1\)) = 300 K
- Final temperature (\(T_2\)) = 500 K
- Change in internal energy (\(\Delta U\)) = 5000 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 change in internal energy for an ideal gas at constant volume is given by:
\[ \Delta U = n \cdot C_v \cdot \Delta T \]
Rearranging this formula to solve for \(C_v\):
\[ C_v = \frac{\Delta U}{n \cdot \Delta T} \]

4. **Substitute the Known Values into the Formula:**
\[ C_v = \frac{5000 \, \text{J}}{4 \, \text{moles} \cdot 200 \, \text{K}} \]

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

### Final Answer:
The molar heat capacity at constant volume (\(C_v\)) is \(6.25 \, \text{J/K/mol}\).

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