First slide

Internal energy

Question

Nitrogen of mass 15 g is enclosed in a vessel at temperature 300 K so that it does not lose any energy to external system. The amount of heat required to double the root mean square velocity of these molecules is N times 2000 J. Find N. (Round off answer to nearest multiple of 10).

Difficult

Solution

Using Law of equipartition of energy,
Energy of a gas at temperature T is $U = \frac{f n R T}{2}$ .
Since, Nitrogen is diatomic, f = 5. Here, $n = \frac{15}{28}$
Also, $v_{R M S} = \sqrt{\frac{3 k T}{m}} α \sqrt{T}$
If $v_{R M S}$ doubles, $\Rightarrow$ T becomes 4 times.
So, $U_{1} = \frac{5 (\frac{5}{28}) (8.3) (300)}{2}$
$U_{2} = \frac{5 (\frac{5}{28}) (8.3) (1200)}{2}$
So, $Δ U = U_{2} - U_{1} = \frac{5 (\frac{15}{28}) (8.3) (1200 - 300)}{2} = 10004.5 J \approx 10000 J = 5 \times 2000 J$

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