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 components and their properties
- We have 3 moles of diatomic oxygen (O2) and 5 moles of monatomic argon (Ar).
- The specific heat at constant volume (Cv) for diatomic gases (like O2) is given by:
\[ C_{v1} = \frac{5}{2} R \]
- The specific heat at constant volume (Cv) for monatomic gases (like Ar) is given by:
\[ C_{v2} = \frac{3}{2} R \]

### Step 2: Write the formula for total internal energy
The total internal energy (U) of the system can be expressed as:
\[ U = U_1 + U_2 \]
Where:
- \( U_1 \) is the internal energy of O2
- \( U_2 \) is the internal energy of Ar

### Step 3: Calculate the internal energy for each gas
Using the formula for internal energy:
\[ U = n C_v T \]
For O2 (3 moles):
\[ U_1 = n_1 C_{v1} T = 3 \cdot \frac{5}{2} R \cdot T = \frac{15}{2} RT \]

For Argon (5 moles):
\[ U_2 = n_2 C_{v2} T = 5 \cdot \frac{3}{2} R \cdot T = \frac{15}{2} RT \]

### Step 4: Combine the internal energies
Now, we can combine the internal energies of both gases:
\[ U = U_1 + U_2 = \frac{15}{2} RT + \frac{15}{2} RT = 15 RT \]

### Step 5: Conclusion
The total internal energy of the system is:
\[ U = 15 RT \]