The internal energy of one mole of the diatomic gas at 200 K is

Options

A. 100 R
B. 200 R
C. 500 R
D. 400 R

Answer

To find the internal energy of one mole of a diatomic gas at 200 K, we can use the formula for internal energy:

### Step-by-Step Solution:

1. **Identify the Formula**:
The internal energy \( U \) of an ideal gas can be expressed as:
\[ U = \frac{F}{2} nRT \]
where:
- \( F \) = degrees of freedom
- \( n \) = number of moles
- \( R \) = universal gas constant (approximately \( 8.314 \, \text{J/(mol K)} \))
- \( T \) = temperature in Kelvin

2. **Determine the Degrees of Freedom**:
For a diatomic gas, the degrees of freedom \( F \) is 5 (3 translational + 2 rotational).

3. **Substitute the Known Values**:
Given:
- \( n = 1 \) mole
- \( T = 200 \, \text{K} \)
- \( R = 8.314 \, \text{J/(mol K)} \)

Substitute these values into the formula:
\[ U = \frac{5}{2} \times 1 \times 8.314 \times 200 \]

4. **Calculate the Internal Energy**:
\[ U = \frac{5}{2} \times 8.314 \times 200 \]
\[ U = 5 \times 8.314 \times 100 \]
\[ U = 4157 \, \text{J} \]

5. **Express in Terms of R**:
Since \( R \approx 8.314 \, \text{J/(mol K)} \), we can express the internal energy in terms of \( R \):
\[ U = 500 R \]

6. **Final Answer**:
Therefore, the internal energy of one mole of the diatomic gas at 200 K is:
\[ U = 500 R \]