The RMS velocity ( $\mathrm{u}_{\mathrm{rms}}$ ) of one mole of an ideal gas was measured at different…

The RMS velocity ( $\mathrm{u}_{\mathrm{rms}}$ ) of one mole of an ideal gas was measured at different temperatures and the following graph is obtained. What is the slope $(\mathrm{m})$ of straight line? $\left(x \text {-axis }=\mathrm{T}(\mathrm{~K}) ; y \text {-axis }=\left(\mathrm{u}_{\mathrm{rms}}\right)^2 ; \mathrm{M}=\text { molar mass } ; \mathrm{R}=\right.\text { gas }$ constant)

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

Solution

RMS velocity $\left(v_{\mathrm{rms}}\right)=\sqrt{\frac{3 \mathrm{RT}}{\mathrm{M}}}$ $\begin{aligned} & \quad\left(v_{\mathrm{rms}}\right)^2=\frac{3 R T}{M} \\ & \therefore \quad\left(v_{\mathrm{rms}}\right)^2=\frac{3 R}{M} \cdot T \\ & y=m x+C \\ & y-\mathrm{axis}=\left(v_{\mathrm{rms}}\right)^2 \\ & \mathrm{x} \text { axis }=\mathrm{T} \\ & \text { Slope }=\frac{3 R}{M} \\ & \text { intercept }(C)=0 \end{aligned}$

Asked in: AP EAMCET 2024 (21 May Shift 1)

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