For a gas sample with N o number of molecules- function N V is given by-N V - d NdV -3 N - V - 3 V 2 for 0 V - where d N is the number of molecules in the speed range V to V - dV- If the rmsspeed of the molecules is nV 0-Then the value of n is

Vrms2= lt;V gt;=V12+V22 +V32+......N=∫V2dN∫dN nbsp;and nbsp;dNdv=N(V)Vrms2=1N∫0∞N(V)V2dV=1N∫0∞3NV∘3.V2V2dV=35V∘2⇒Vrms=35V∘ nbsp; √(n) = √(%s/%s)35 ∴ ...

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Byju's Answer

Standard XII

Chemistry

Dynamic Nature of Physical and Chemical Equilibrium

For a gas sam...

Question

For a gas sample with $N_{\circ}$ number of molecules, function N(V) is given by :
$N (V) = \frac{d N}{d V} = (\frac{3 N_{\circ}}{V_{\circ}^{3}}) V^{2} f o r 0 < V < V_{\circ} a n d N (V) = 0 f o r V > V_{\circ}$ where dN is the number of molecules in the speed range $V t o V + d V$ . If the rms speed of the molecules is $\sqrt{n} V_{0}$ .Then the value of $n$ is

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Solution

$V_{r m s}^{2} = < V > = \frac{V_{1}^{2} + V_{2}^{2} + V_{3}^{2} + . . . . . .}{N} = \frac{\int V^{2} d N}{\int d N} a n d \frac{d N}{d v} = N (V) V_{r m s}^{2} = \frac{1}{N} \int_{0}^{\infty} N (V) V^{2} d V = \frac{1}{N} \int_{0}^{\infty} (\frac{3 N}{V_{\circ}^{3}} . V^{2}) V^{2} d V = \frac{3}{5} V_{\circ}^{2} \Rightarrow V_{r m s} = \sqrt{\frac{3}{5}} V_{\circ}$
$⟹$ $\sqrt{n}$ = $\sqrt{\frac{3}{5}}$
$∴$ $n$ = 0.6

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For a gas sample with N∘ number of molecules, function N(V) is given by : N(V)=dNdV=(3N∘V3∘)V2 for 0<V<V∘and N(V)=0 for V>V∘where dN is the number of molecules in the speed range V to V+dV. If the rms speed of the molecules is √nV0.Then the value of n is

V2rms=<V>=V21+V22+V23+......N=∫V2dN∫dN and dNdv=N(V)V2rms=1N∫∞0N(V)V2dV=1N∫∞0(3NV3∘.V2)V2dV=35V2∘⇒Vrms=√35V∘ ⟹ √n = √35 ∴ n = 0.6