Find total internal energy of 3 moles of hydrogen gas at temperature `T`.

Explanation

To find the total internal energy of 3 moles of hydrogen gas at temperature \( T \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Formula for Internal Energy**:
The internal energy \( U \) of an ideal gas can be expressed as:
\[ U = n C_V T \]
where:
- \( n \) = number of moles,
- \( C_V \) = molar heat capacity at constant volume,
- \( T \) = temperature.

2. **Determine the Number of Moles**:
From the problem, we know:
\[ n = 3 \text{ moles} \]

3. **Identify the Type of Gas**:
The gas in question is hydrogen (\( H_2 \)), which is a diatomic gas.

4. **Calculate the Degrees of Freedom**:
For a diatomic gas, the degrees of freedom \( F \) is given by:
\[ F = 5 \]
(3 translational + 2 rotational).

5. **Calculate \( C_V \)**:
The molar heat capacity at constant volume \( C_V \) for a diatomic gas is given by:
\[ C_V = \frac{F}{2} R = \frac{5}{2} R \]
where \( R \) is the universal gas constant.

6. **Substitute \( C_V \) into the Internal Energy Formula**:
Now, substituting \( C_V \) into the internal energy formula:
\[ U = n C_V T = n \left(\frac{5}{2} R\right) T \]

7. **Insert the Value of \( n \)**:
Substituting \( n = 3 \):
\[ U = 3 \left(\frac{5}{2} R\right) T \]

8. **Simplify the Expression**:
\[ U = \frac{15}{2} R T \]

9. **Final Result**:
Thus, the total internal energy of 3 moles of hydrogen gas at temperature \( T \) is:
\[ U = \frac{15}{2} R T \]