An ideal gas in a container of volume 500 c.c. is at a pressure of $2 \times 10^{+5} \mathrm{~N} /…

An ideal gas in a container of volume 500 c.c. is at a pressure of $2 \times 10^{+5} \mathrm{~N} / \mathrm{m}^2$. The average kinetic energy of each molecule is $6 \times 10^{-21} \mathrm{~J}$. The number of gas molecules in the container is
  1. $5 \times 10^{25}$
  2. $25 \times 10^{23}$
  3. $5 \times 10^{23}$
  4. $2.5 \times 10^{22}$

Solution

$\begin{aligned} & \mathrm{PV}=\mathrm{Nk}_{\mathrm{B}} \mathrm{T} \\ & \text { and, K.E./molecule }=\frac{3}{2} \mathrm{k}_{\mathrm{B}} \mathrm{T}=\frac{3}{2} \frac{\mathrm{PV}}{\mathrm{N}}\end{aligned}$ $\begin{aligned} \therefore \quad \mathrm{N} & =\frac{3}{2} \times \frac{\mathrm{PV}}{(\mathrm{K} . \mathrm{E} . / \mathrm{molecule})} \\ & =\frac{3}{2} \times \frac{2 \times 10^5 \times 500 \times 10^{-6}}{6 \times 10^{-21}}=2.5 \times 10^{22}\end{aligned}$

Asked in: MHT CET 2023 (09 May Shift 2)

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