iIn isochoric process change ib internal energy of the gas in `triangle` T temperature rise for 2 mole is

Options

A. `2C_p/_\T`
B. `C_V/2/_\T`
C. `2C_V/_\T`
D. Zero

Answer

To solve the problem of finding the change in internal energy (ΔU) of a gas during an isochoric process for 2 moles with a temperature rise of ΔT, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Isochoric Process**:
In an isochoric process, the volume of the gas remains constant. Therefore, any heat added to the system increases the internal energy of the gas.

2. **Use the Formula for Change in Internal Energy**:
The change in internal energy (ΔU) for an ideal gas can be expressed as:
\[ \Delta U = n \cdot C_v \cdot \Delta T \]
where:
- \( n \) = number of moles of the gas
- \( C_v \) = molar specific heat at constant volume
- \( \Delta T \) = change in temperature

3. **Identify Given Values**:
From the question, we know:
- Number of moles, \( n = 2 \)
- Change in temperature, \( \Delta T \) (as given in the question)

4. **Relate \( C_v \) to Degrees of Freedom**:
The molar specific heat at constant volume \( C_v \) can be expressed in terms of the degrees of freedom \( F \) of the gas:
\[ C_v = \frac{F}{2} R \]
where \( R \) is the universal gas constant.

5. **Substitute \( C_v \) into the ΔU Formula**:
Now, substituting the expression for \( C_v \) into the equation for ΔU:
\[ \Delta U = n \cdot \left(\frac{F}{2} R\right) \cdot \Delta T \]
Substituting \( n = 2 \):
\[ \Delta U = 2 \cdot \left(\frac{F}{2} R\right) \cdot \Delta T \]
Simplifying this gives:
\[ \Delta U = F \cdot R \cdot \Delta T \]

6. **Final Expression**:
The final expression for the change in internal energy for 2 moles of gas during an isochoric process with a temperature rise of ΔT is:
\[ \Delta U = F \cdot R \cdot \Delta T \]