3 mole of O2 mixed with 5 moles of Argon at temperature T. Find total internal energy of system

Options

A. 12RT
B. 19RT
C. 15RT
D. 10RT

Answer

To find the total internal energy of a system consisting of 3 moles of O2 mixed with 5 moles of Argon at temperature T, we can follow these steps:

### Step 1: Identify the degrees of freedom for each gas
- For diatomic gases like O2, the degrees of freedom (F) is 5.
- For monatomic gases like Argon, the degrees of freedom (F) is 3.

### Step 2: Write down the formula for internal energy
The total internal energy (U) of a gas can be calculated using the formula:
\[ U = \frac{F_1}{2} N_1 R T + \frac{F_2}{2} N_2 R T \]
where:
- \( F_1 \) = degrees of freedom of gas 1 (O2)
- \( N_1 \) = number of moles of gas 1 (O2)
- \( F_2 \) = degrees of freedom of gas 2 (Argon)
- \( N_2 \) = number of moles of gas 2 (Argon)
- \( R \) = universal gas constant
- \( T \) = temperature

### Step 3: Substitute the values into the formula
- For O2 (gas 1):
- \( F_1 = 5 \)
- \( N_1 = 3 \)

- For Argon (gas 2):
- \( F_2 = 3 \)
- \( N_2 = 5 \)

Now substituting these values into the formula:
\[ U = \frac{5}{2} \times 3 \times R T + \frac{3}{2} \times 5 \times R T \]

### Step 4: Calculate each term
- For O2:
\[ U_1 = \frac{5}{2} \times 3 \times R T = \frac{15}{2} R T \]

- For Argon:
\[ U_2 = \frac{3}{2} \times 5 \times R T = \frac{15}{2} R T \]

### Step 5: Add the internal energies of both gases
\[ U = U_1 + U_2 = \frac{15}{2} R T + \frac{15}{2} R T = \frac{30}{2} R T = 15 R T \]

### Final Answer
The total internal energy of the system is:
\[ U = 15 R T \]

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