If c is r.m.s. speed of molecules in a gas and v is the speed of sound waves in the gas, show that c/v is constant and independent of temperature for all diatomic gases.

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Payal Gupta | Contributor-Level 10

3 months ago

This is a long answer type question as classified in NCERT Exemplar

we know that rms speed of molecules of a gas

C= $\sqrt[]{\frac{3 p}{ρ}} = \sqrt[]{\frac{3 R T}{M}}$

Where M is the molar mass of the gas

Speed of sound wave in gas v= $\sqrt[]{\frac{γ p}{ρ}} = \sqrt[]{\frac{γ R T}{M}}$

On dividing above equation we get

c/v = $\sqrt[]{\frac{3 R T}{M} \times \frac{M}{γ R T}}$

c/v = $\sqrt[]{\frac{3}{γ}}$

where $γ$ = adiabatic constant for diatomic gas

$γ = \frac{7}{5}$

c/v=constant

This is a long answer type question as classified in NCERT Exemplarwe know that rms speed of molecules of a gasC= <math> <mroot> <mrow> <mrow> <mfrac> <mrow> <mrow> <mn>3</mn> <mi>p</mi> </mrow> </mrow> <mrow> <mrow> <mi>ρ</mi> </mrow> </mrow> </mfrac> </mrow> </mrow> <mrow></mrow> </mroot> <mo>=</mo> <mroot> <mrow> <mrow> <mfrac> <mrow> <mrow> <mn>3</mn> <mi>R</mi> <mi>T</mi> </mrow> </mrow> <mrow> <mrow> <mi>M</mi> </mrow> </mrow> </mfrac> </mrow> </mrow> <mrow></mrow> </mroot> <mi> </mi> <mi> </mi> </math>  Where M is the molar mass of the gasSpeed of sound wave in gas v= <math> <mroot> <mrow> <mrow> <mfrac> <mrow> <mrow> <mi>γ</mi> <mi>p</mi> </mrow> </mrow> <mrow> <mrow> <mi>ρ</mi> </mrow> </mrow> </mfrac> </mrow> </mrow> <mrow></mrow> </mroot> <mo>=</mo> <mroot> <mrow> <mrow> <mfrac> <mrow> <mrow> <mi>γ</mi> <mi>R</mi> <mi>T</mi> </mrow> </mrow> <mrow> <mrow> <mi>M</mi> </mrow> </mrow> </mfrac> </mrow> </mrow> <mrow></mrow> </mroot> </math> On dividing above equation we getc/v = <math> <mroot> <mrow> <mrow> <mfrac> <mrow> <mrow> <mn>3</mn> <mi>R</mi> <mi>T</mi> </mrow> </mrow> <mrow> <mrow> <mi>M</mi> </mrow> </mrow> </mfrac> <mo>×</mo> <mfrac> <mrow> <mrow> <mi>M</mi> </mrow> </mrow> <mrow> <mrow> <mi>γ</mi> <mi>R</mi> <mi>T</mi> </mrow> </mrow> </mfrac> </mrow> </mrow> <mrow></mrow> </mroot> </math> c/v = <math> <mroot> <mrow> <mrow> <mfrac> <mrow> <mrow> <mn>3</mn> </mrow> </mrow> <mrow> <mrow> <mi>γ</mi> </mrow> </mrow> </mfrac> </mrow> </mrow> <mrow></mrow> </mroot> </math> where <math> <mi>γ</mi> </math> = adiabatic constant for diatomic gas <math> <mi>γ</mi> <mo>=</mo> <mfrac> <mrow> <mrow> <mn>7</mn> </mrow> </mrow> <mrow> <mrow> <mn>5</mn> </mrow> </mrow> </mfrac> </math> c/v=constant

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If c is r.m.s. speed of molecules in a gas and v is the speed of sound waves in the gas, show that c/v is constant and independent of temperature for all diatomic gases.

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