Molar heat capacity of an ideal gas at constant volume is given by `C_(V)=2xx10^(-2)J` (in Joule). If 3.5 mole of hits ideal gas are heated at constant volume from 300K to 400K the change in internal energy will be

Options

A. 7
B. 4
C. 8
D. 6

Explanation

To solve the problem, we need to calculate the change in internal energy (\( \Delta U \)) of the ideal gas using the formula:

\[ \Delta U = n \cdot C_V \cdot \Delta T \]

Where:
- \( n \) = number of moles of the gas
- \( C_V \) = molar heat capacity at constant volume
- \( \Delta T \) = change in temperature

### Step 1: Identify the given values
- Molar heat capacity \( C_V = 2 \times 10^{-2} \, \text{J/mol K} \)
- Number of moles \( n = 3.5 \, \text{moles} \)
- Initial temperature \( T_1 = 300 \, \text{K} \)
- Final temperature \( T_2 = 400 \, \text{K} \)

### Step 2: Calculate the change in temperature (\( \Delta T \))
\[ \Delta T = T_2 - T_1 = 400 \, \text{K} - 300 \, \text{K} = 100 \, \text{K} \]

### Step 3: Substitute the values into the formula for \( \Delta U \)
Now we can substitute the values into the formula:
\[ \Delta U = n \cdot C_V \cdot \Delta T \]
\[ \Delta U = 3.5 \, \text{moles} \cdot (2 \times 10^{-2} \, \text{J/mol K}) \cdot (100 \, \text{K}) \]

### Step 4: Perform the multiplication
Calculating the multiplication step by step:
1. First, calculate \( C_V \cdot \Delta T \):
\[ C_V \cdot \Delta T = (2 \times 10^{-2} \, \text{J/mol K}) \cdot (100 \, \text{K}) = 2 \, \text{J/mol} \]

2. Now multiply by the number of moles:
\[ \Delta U = 3.5 \, \text{moles} \cdot 2 \, \text{J/mol} = 7 \, \text{J} \]

### Step 5: Final answer
Thus, the change in internal energy is:
\[ \Delta U = 7 \, \text{J} \]