Question

A vessel contains \(3\) moles of He, \(1\) mole of Ar, \(5\) moles of \(N_2\), and \(3\) moles of \(H_2\). If the vibrational modes are ignored, the total internal energy of the system of gases is

• A. \(20RT\)
• B. \(26RT\)
• C. \(25RT\)
• D. \(30RT\)

Hint:

For ideal gases, \[ U=\frac{f}{2}nRT. \] Monoatomic gases have \[ f=3, \] while diatomic gases without vibrational modes have \[ f=5. \]

Answer 1

The Correct Option is B

Step 1: Assign degrees of freedom (no vibrational modes).
He (monatomic): \( f=3 \); Ar (monatomic): \( f=3 \); N2 (diatomic): \( f=5 \); H2 (diatomic): \( f=5 \).

Step 2: Total internal energy \( U = \sum n_i\frac{f_i}{2}RT \).
\( U = \left(3\times\frac{3}{2} + 1\times\frac{3}{2} + 5\times\frac{5}{2} + 3\times\frac{5}{2}\right)RT \)
\( = \left(\frac{9}{2}+\frac{3}{2}+\frac{25}{2}+\frac{15}{2}\right)RT = \frac{52}{2}RT = 26RT \)

\[ \boxed{U = 26RT} \]