At $\mathrm{T}(\mathrm{K})$, the $\mathrm{P}, \mathrm{V}$ and $u_{\mathrm{rms}}$ of 1 mole of an ideal gas…

At $\mathrm{T}(\mathrm{K})$, the $\mathrm{P}, \mathrm{V}$ and $u_{\mathrm{rms}}$ of 1 mole of an ideal gas were measured. The following graph is obtained. What is its slope (m)? $\left(x\right.$-axis $=\mathrm{PV} ; y$-axis $u_{r m s}^2, \mathrm{M}=$ Molar mass $)$

1. $\frac{3}{M}$
2. $\frac{M}{3}$
3. $\left(\frac{M}{3}\right)^{1 / 2}$
4. $\left(\frac{3}{M}\right)^{1 / 2}$

Solution

Root mean square velocity $\left(v_{\text {rms }}\right)=\sqrt{\frac{3 \mathrm{RT}}{\mathrm{M}}}$ for 1 mole ideal gas equation $\begin{gathered} \mathrm{PV}=\mathrm{RT} \\ \mathrm{v}_{\mathrm{rms}}=\sqrt{\frac{3 \mathrm{PV}}{\mathrm{M}}} \\ \mathrm{v}_{\mathrm{rms}}^2=\frac{3 \mathrm{PV}}{\mathrm{M}} \end{gathered}$
$\begin{aligned} & \mathrm{y}=\mathrm{mx}+\mathrm{C} \\ & \mathrm{M}(\text { slope })=\frac{3}{\mathrm{M}} \\ & \text { Intercept }(C)=0\end{aligned}$

Asked in: AP EAMCET 2024 (21 May Shift 2)

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