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 the internal energy of an ideal gas:

\[ U = \frac{f}{2} RT \]

Where:
- \( U \) is the internal energy,
- \( f \) is the degrees of freedom,
- \( R \) is the universal gas constant (approximately \( 8.314 \, \text{J/(mol K)} \)),
- \( T \) is the temperature in Kelvin.

### Step 1: Identify the degrees of freedom for a diatomic gas
For a diatomic gas, the degrees of freedom \( f \) is typically 5 at low temperatures (where vibrational modes are not excited). This is because:
- 2 translational degrees of freedom (movement in x and y directions),
- 2 rotational degrees of freedom (rotation about two axes),
- 1 vibrational degree of freedom is not considered at low temperatures.

Thus, for our calculation:
\[ f = 5 \]

### Step 2: Substitute the values into the internal energy formula
Now we can substitute the values into the internal energy formula:
\[ U = \frac{5}{2} RT \]
Substituting \( R = 8.314 \, \text{J/(mol K)} \) and \( T = 200 \, \text{K} \):
\[ U = \frac{5}{2} \times 8.314 \times 200 \]

### Step 3: Calculate the internal energy
First, calculate \( R \times T \):
\[ RT = 8.314 \times 200 = 1662.8 \, \text{J/mol} \]
Now, substitute this back into the equation for \( U \):
\[ U = \frac{5}{2} \times 1662.8 \]
Calculating this gives:
\[ U = \frac{5 \times 1662.8}{2} = \frac{8314}{2} = 4157 \, \text{J/mol} \]

### Final Answer
The internal energy of one mole of the diatomic gas at 200 K is:
\[ U = 4157 \, \text{J} \]