According to kinetic theory of gases,

A. The motion of the gas molecules freezes at 0°C.

B. The mean free path of gas molecules decreases if the density of molecules is increased.

C. The mean free path of gas molecules increases if temperature is increased keeping pressure constant.

D. Average kinetic energy per molecules per degree of freedom is $\frac{3}{2} k_{B} T$ (for monoatomic gases).

Choose the most appropriate answer form the options given below

Option 1 -

A and C only

Option 2 -

B and C only

Option 3 -

A and B only

Option 4 -

C and D only

3 Views | Posted 3 months ago

Asked by Shiksha User

1 Answer

Answered by

Payal Gupta | Contributor-Level 10

3 months ago

Correct Option - 2

Detailed Solution:

$\bar{x}$ (mean free path) = $\frac{1}{\sqrt[]{2} π d^{2} n_{v}}$

$n_{v} = \frac{n N_{A}}{v},$ n → No. of moles in volume v N_A ® Avogadro’s Number

$\frac{n}{v} = \frac{P}{R_{T}}$

$\bar{x} \propto v \propto \frac{1}{ρ (d e n s i t y} & \bar{x} α T a t c o n s t a n t P)$

^{$ρ ↑ \Leftrightarrow \bar{x} ↓ | \bar{x} ↑ \Leftrightarrow T ↑$}

The motion of the gas molecules freezes at 0K not 0° C

Average kinetic Energy per molecule per degree of freedom is $= \frac{1}{2} k_{B} T$ (for Mono atomic gases)

<math> <mrow> <mover accent="true"> <mi>x</mi> <mo>¯</mo> </mover> </mrow> </math> (mean free path) = <math> <mrow> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mroot> <mrow> <mn>2</mn> </mrow> <mrow></mrow> </mroot> <mi>π</mi> <msup> <mrow> <mi>d</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <msub> <mrow> <mi>n</mi> </mrow> <mrow> <mi>v</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </math>   <math> <mrow> <msub> <mrow> <mi>n</mi> </mrow> <mrow> <mi>v</mi> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <mi>n</mi> <msub> <mrow> <mi>N</mi> </mrow> <mrow> <mi>A</mi> </mrow> </msub> </mrow> <mrow> <mi>v</mi> </mrow> </mfrac> <mo>,</mo> </mrow> </math> n → No. of moles in volume v NA ® Avogadro’s Number <math> <mrow> <mfrac> <mrow> <mi>n</mi> </mrow> <mrow> <mi>v</mi> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mi>P</mi> </mrow> <mrow> <msub> <mrow> <mi>R</mi> </mrow> <mrow> <mi>T</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </math> <math> <mrow> <mover accent="true"> <mi>x</mi> <mo>¯</mo> </mover> <mo>∝</mo> <mi>v</mi> <mo>∝</mo> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mi>ρ</mi> <mo stretchy="false">(</mo> <mi>d</mi> <mi>e</mi> <mi>n</mi> <mi>s</mi> <mi>i</mi> <mi>t</mi> <mi>y</mi> </mrow> </mfrac> <mo>&</mo> <mtext> </mtext> <mtext> </mtext> <mover accent="true"> <mi>x</mi> <mo>¯</mo> </mover> <mi>α</mi> <mi>T</mi> <mtext> </mtext> <mtext> </mtext> <mi>a</mi> <mi>t</mi> <mtext> </mtext> <mi>c</mi> <mi>o</mi> <mi>n</mi> <mi>s</mi> <mi>t</mi> <mi>a</mi> <mi>n</mi> <mi>t</mi> <mtext> </mtext> <mi>P</mi> <mo stretchy="false">)</mo> </mrow> </math> <math> <mrow> <mi>ρ</mi> <mo>↑</mo> <mo>⇔</mo> <mover accent="true"> <mi>x</mi> <mo>¯</mo> </mover> <mo>↓</mo> <mtext> </mtext> <mtext> </mtext> <mo>|</mo> <mover accent="true"> <mi>x</mi> <mo>¯</mo> </mover> <mo>↑</mo> <mo>⇔</mo> <mi>T</mi> <mo>↑</mo> </mrow> </math>                           The motion of the gas molecules freezes at 0K not 0° CAverage kinetic Energy per molecule per degree of freedom is  <math> <mrow> <mo>=</mo> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> <msub> <mrow> <mi>k</mi> </mrow> <mrow> <mi>B</mi> </mrow> </msub> <mi>T</mi> </mrow> </math> (for Mono atomic gases)

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