Heat and Thermodynamics 4 Question 11
13. A gas mixture consists of 3 moles of oxygen and 5 moles of argon at temperature $T$. Considering only translational and rotational modes, the total internal energy of the system is
(2019 Main, 11 Jan I)
(a) $12 R T$
(b) $15 R T$
(c) $20 R T$
(d) $4 R T$
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Solution:
- Internal energy of a gas with $f$ degree of freedom is
$U=\frac{n f R T}{2}$, where $n$ is the number of moles.
Internal energy due to $O _2$ gas which is a diatomic gas is
$U _1=\frac{n _1 f _1 R T}{2}=3 \times \frac{5}{2} R T$
$\left(\because n _1=3\right.$ moles, degree of freedom for a diatomic gas $f _1=5$ )
Internal energy due to Ar gas which is a monoatomic gas is
$$ U _2=\frac{n _2 f _2 R T}{2}=5 \times \frac{3}{2} R T $$
$\left(\because n _2=5\right.$ moles, degree of freedom for a monoatomic gas $f _2=3$ )
$\therefore$ Total internal energy $=U _1+U _2$
$$ \Rightarrow \quad U=15 R T $$
