The graph shows hypothetical speed distribution for a sample of - gas particles for V0- dNdV-0

a) d N/d V→ v d N/d v=a/v_0· v av_0=2 (Area under the curve) Area= ∫ ydx=∫dN/dv.dv=∫ dN=N b) d N/d v=a v/v_0 d Na v/v_0 d v v_a v g=∫ d N v/∫ d N v ...

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Byju's Answer

Standard XII

Physics

Pressure vs Velocity

The graph sho...

Question

The graph shows hypothetical speed distribution for a sample of 𝑁 gas particles (for $V_{0}, \frac{d N}{d V} = 0$ )

The value of

a V_{0}

2 N

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The ratio

\frac{V_{a v g}}{V_{0}}

is equal to

\frac{2}{3}

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The ratio

\frac{V_{r m s}}{V_{0}}

is equal to

\frac{1}{\sqrt{2}}

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Three-fourths of the total particles have a speed between

0.5 V_{0}

and

V_{0}

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Solution

The correct option is D Three-fourths of the total particles have a speed between $0.5 V_{0}$ and $V_{0}$

a)

$\frac{d N}{d V} \to v \frac{d N}{d v} = \frac{a}{v_{0}} \cdot v$

$a v_{0} = 2$ (Area under the curve)

Area= $\int y d x = \int \frac{d N}{d v} . d v = \int d N = N$

b)

$\frac{d N}{d v} = \frac{a v}{v_{0}}$

$d N \frac{a v}{v_{0}} d v$

$v_{a v g} = \frac{\int d N v}{\int d N}$

$v_{a v g} = \frac{\int_{0}^{v_{0}} \frac{a v}{v_{0}} d v v}{v_{0} \int \frac{a v}{v_{0}} d v}$

$v_{a v g} = \frac{\frac{a}{v_{0}} \int_{0}^{v_{0}} v^{2} d v}{\frac{a}{v_{0}} v_{0}^{v_{0}} v d v} = \frac{2 V_{0}}{3}$

c)

$v_{r m s} = \sqrt{m e a n s q}$

$m e a n s q = \frac{\int d N v^{2}}{\int d N}$

$m e a n s q = \frac{\int_{0}^{V_{0}} \frac{a v}{v_{0}} d v v^{2}}{\int_{0}^{V_{0}} \frac{a v}{v_{0}} d v}$

$\frac{\int_{0}^{V_{0}} v^{3} d v}{\int_{0}^{V_{0}} v d v} = \frac{v_{0}^{4}}{4 \frac{V_{0}^{2}}{4}} = 0.5 V_{0}^{2}$

$\begin{matrix} v_{r m s} & = \sqrt{\frac{1}{2} v_{0}^{2}} V_{r m s} & = \frac{v_{0}}{\sqrt{2}} \frac{V_{r m s}}{v_{0}} & = \frac{1}{\sqrt{2}} \end{matrix}$

$\frac{v_{0}}{2} + v_{0} \to \frac{3}{4}$ Number of molequles.

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The graph shows hypothetical speed distribution for a sample of 𝑁 gas particles (for V0, dNdV=0)

The graph shows hypothetical speed distribution for a sample of 𝑁 gas particles (for $V_{0}, \frac{d N}{d V} = 0$ )