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 problem, we can follow these steps:

### Step 1: Identify the given values
- Number of moles (n) = 4 moles
- Initial temperature (T_initial) = 300 K
- Final temperature (T_final) = 500 K
- Change in internal energy (ΔU) = 5000 J

### Step 2: Calculate the change in temperature (ΔT)
\[ \Delta T = T_{final} - T_{initial} = 500 \, K - 300 \, K = 200 \, K \]

### Step 3: Use the formula for change in internal energy at constant volume
The change in internal energy (ΔU) for an ideal gas at constant volume can be expressed as:
\[ \Delta U = n \cdot C_V \cdot \Delta T \]

### Step 4: Rearrange the formula to solve for C_V
To find the molar heat capacity at constant volume (C_V), rearrange the equation:
\[ C_V = \frac{\Delta U}{n \cdot \Delta T} \]

### Step 5: Substitute the known values into the equation
Substituting the values we have:
\[ C_V = \frac{5000 \, J}{4 \, moles \cdot 200 \, K} \]

### Step 6: Calculate C_V
\[ C_V = \frac{5000}{800} = 6.25 \, \frac{J}{mole \cdot K} \]

### Final Answer
The molar heat capacity at constant volume (C_V) is **6.25 J/(mole·K)**.

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