KCET · Physics · Thermodynamics
A gas mixture contains monoatomic and diatomic molecules of 2 moles each. The mixture has a total internal energy of (symbols have usual meanings)
- A \(3 R T\)
- B \(5 R T\)
- C \(8 R T\)
- D \(9 R T\)
Answer & Solution
Correct Answer
(C) \(8 R T\)
Step-by-step Solution
Detailed explanation
Total internal energy of a gas is given as
\(U=\frac{n}{2} f R T\)
where, \(n=\) number of moles
and \(f=\) degree of freedom.
Given, \(n_{\text {diatomic }}=n_{\text {monoatomic }}=2\)
As, \(f_{\text {monoatomic }}=3\)
\(f_{\text {diatomic }} =5 \)
\( \Rightarrow U_{\text {monoatomic }} =\frac{2}{2} \times 3 R T=3 R T \)
\( U_{\text {diatomic }} =\frac{2}{2} \times 5 R T=5 R T\)
\(\therefore\) Total internal energy of the mixture of gases,
\(\begin{aligned}
U_{\text {total }} &=U_{\text {monoatomic }}+U_{\text {diatomic }} \\
&=3 R T+5 R T=8 R T
\end{aligned}\)
\(U=\frac{n}{2} f R T\)
where, \(n=\) number of moles
and \(f=\) degree of freedom.
Given, \(n_{\text {diatomic }}=n_{\text {monoatomic }}=2\)
As, \(f_{\text {monoatomic }}=3\)
\(f_{\text {diatomic }} =5 \)
\( \Rightarrow U_{\text {monoatomic }} =\frac{2}{2} \times 3 R T=3 R T \)
\( U_{\text {diatomic }} =\frac{2}{2} \times 5 R T=5 R T\)
\(\therefore\) Total internal energy of the mixture of gases,
\(\begin{aligned}
U_{\text {total }} &=U_{\text {monoatomic }}+U_{\text {diatomic }} \\
&=3 R T+5 R T=8 R T
\end{aligned}\)
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