Question

The total internal energy of 2 moles of a monoatomic gas at a temperature 27°C is \( U \). The total internal energy of 3 moles of a diatomic gas at a temperature 127°C is:

• A. \( U \)
• B. \( \frac{10U}{3} \)
• C. \( 2U \)
• D. \( 3U \)

Hint:

The internal energy of a gas depends on the number of moles, degrees of freedom, and absolute temperature. Always use \( U = \frac{f}{2} nRT \) to compute internal energy.

Answer 1

The Correct Option is B

Step 1: Understanding the Internal Energy Formula
The internal energy of an ideal gas is given by: \[ U = \frac{f}{2} nRT \] where: - \( f \) is the degrees of freedom,
- \( n \) is the number of moles,
- \( R \) is the universal gas constant,
- \( T \) is the absolute temperature.
Step 2: Internal Energy for Monoatomic Gas
For a monoatomic gas (\( f = 3 \)): \[ U_1 = \frac{3}{2} (2R T_1) \] Given that at \( T_1 = 27^\circ C = 300K \), the internal energy is: \[ U_1 = U \] Step 3: Internal Energy for Diatomic Gas For a diatomic gas (\( f = 5 \)): \[ U_2 = \frac{5}{2} (3 R T_2) \] Given that \( T_2 = 127^\circ C = 400K \), we calculate: \[ U_2 = \frac{5}{2} (3 R \times 400) \] \[ = \frac{5}{2} \times 3 \times \frac{U}{\frac{3}{2} \times 2 \times 300} \] \[ = \frac{5}{2} \times 3 \times \frac{U}{3 \times 300} \] \[ = \frac{5}{2} \times \frac{U}{2} \times \frac{400}{300} \] \[ = \frac{5}{2} \times \frac{U}{2} \times \frac{4}{3} \] Step 4: Conclusion \[ U_2 = \frac{10U}{3} \] Thus, the correct answer is option (B) \( \frac{10U}{3} \).