Under an adiabatic process, the volume of an ideal gas gets doubled. Consequently the mean collision time between the gas molecule changes from <math> <msub> <mrow> <mrow> <mi>τ</mi> </mrow> </mrow> <mrow> <mrow> <mn>1</mn> </mrow> </mrow> </msub> </math> to <math> <msub> <mrow> <mrow> <mi>τ</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msub> </math> . If <math> <mfrac> <mrow> <mrow> <msub> <mrow> <mrow> <mi>C</mi> </mrow> </mrow> <mrow> <mrow> <mi>p</mi> </mrow> </mrow> </msub> </mrow> </mrow> <mrow> <mrow> <msub> <mrow> <mrow> <mi>C</mi> </mrow> </mrow> <mrow> <mrow> <mi>v</mi> </mrow> </mrow> </msub> </mrow> </mrow> </mfrac> <mo>=</mo> <mi>γ</mi> </math> for the gas then a good estimate for <math> <mfrac> <mrow> <mrow> <msub> <mrow> <mrow> <mi>τ</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msub> </mrow> </mrow> <mrow> <mrow> <msub> <mrow> <mrow> <mi>τ</mi> </mrow> </mrow> <mrow> <mrow> <mn>1</mn> </mrow> </mrow> </msub> </mrow> </mrow> </mfrac> </math> is given by

Relaxation time ( τ ) ∝ V T <!- [endif]-><!- [endif]->and T ∝ 1 V γ - 1 <!- [endif]-><!- [endif]-> τ ∝ V 1 + γ - 1 2 τ ∝ V 1 + γ 2 τ f τ i = 2 V V 1 + γ 2 τ f τ i = ( 2 ) 1 + γ 2 <!- [endif]-><!- [endif]->

Under an adiabatic process, the volume of an ideal gas gets doubled. Consequently the mean collision time between the gas molecule changes from <math> <msub> <mrow> <mrow> <mi>τ</mi> </mrow> </mrow> <mrow> <mrow> <mn>1</mn> </mrow> </mrow> </msub> </math> to <math> <msub> <mrow> <mrow> <mi>τ</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msub> </math> . If <math> <mfrac> <mrow> <mrow> <msub> <mrow> <mrow> <mi>C</mi> </mrow> </mrow> <mrow> <mrow> <mi>p</mi> </mrow> </mrow> </msub> </mrow> </mrow> <mrow> <mrow> <msub> <mrow> <mrow> <mi>C</mi> </mrow> </mrow> <mrow> <mrow> <mi>v</mi> </mrow> </mrow> </msub> </mrow> </mrow> </mfrac> <mo>=</mo> <mi>γ</mi> </math> for the gas then a good estimate for <math> <mfrac> <mrow> <mrow> <msub> <mrow> <mrow> <mi>τ</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msub> </mrow> </mrow> <mrow> <mrow> <msub> <mrow> <mrow> <mi>τ</mi> </mrow> </mrow> <mrow> <mrow> <mn>1</mn> </mrow> </mrow> </msub> </mrow> </mrow> </mfrac> </math> &nbsp;is given by &lt;!-- [if !supportLineBreakNewLine]--&gt; &lt;!--[endif]--&gt;

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Physics Thermodynamics Physics Class 11th Thermodynamics

Under an adiabatic process, the volume of an ideal gas gets doubled. Consequently the mean collision time between the gas molecule changes from $τ_{1}$ to $τ_{2}$ . If $\frac{C_{p}}{C_{v}} = γ$ for the gas then a good estimate for $\frac{τ_{2}}{τ_{1}}$ is given by

Option 1 -

Option 2 -

${(\frac{1}{2})}^{\frac{γ + 1}{2}}$

Option 3 -

${(\frac{1}{2})}^{γ}$

Option 4 -

$\frac{1}{2}$

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1 Answer

Answered by

alok kumar singh | Contributor-Level 10

a month ago

Correct Option - 1

Detailed Solution:

Relaxation time $(τ) \propto \frac{V}{\sqrt[]{T}}$

and $T \propto \frac{1}{V^{γ - 1}}$

$\begin{array}{r} τ \propto V^{1 + \frac{γ - 1}{2}} τ \propto V^{\frac{1 + γ}{2}} \\ \frac{τ_{f}}{τ_{i}} = {(\frac{2 V}{V})}^{\frac{1 + γ}{2}} \frac{τ_{f}}{τ_{i}} = (2)^{\frac{1 + γ}{2}} \end{array}$

Relaxation time <math> <mo> (</mo> <mi>τ</mi> <mo>)</mo> <mo>∝</mo> <mfrac> <mrow> <mrow> <mi mathvariant="normal">V</mi> </mrow> </mrow> <mrow> <mrow> <mroot> <mrow> <mrow> <mi mathvariant="normal">T</mi> </mrow> </mrow> <mrow></mrow> </mroot> </mrow> </mrow> </mfrac> </math> <!- [endif]-><!- [endif]->and <math> <mrow> <mi mathvariant="normal"> </mi> </mrow> <mi mathvariant="normal">T</mi> <mo>∝</mo> <mfrac> <mrow> <mrow> <mn>1</mn> </mrow> </mrow> <mrow> <mrow> <msup> <mrow> <mrow> <mtext> </mtext> <mi mathvariant="normal">V</mi> </mrow> </mrow> <mrow> <mrow> <mi>γ</mi> <mo>-</mo> <mn>1</mn> </mrow> </mrow> </msup> </mrow> </mrow> </mfrac> </math> <!- [endif]-><!- [endif]-> <math> <mtable columnalign="right"> <mtr> <mtd> <mrow></mrow> </mtd> <mtd> <mrow> <mi>τ</mi> <mo>∝</mo> <msup> <mrow> <mrow> <mi mathvariant="normal">V</mi> </mrow> </mrow> <mrow> <mrow> <mn>1</mn> <mo>+</mo> <mfrac> <mrow> <mrow> <mi>γ</mi> <mo>-</mo> <mn>1</mn> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </mfrac> </mrow> </mrow> </msup> <mo></mo> <mrow> <mi mathvariant="normal"> </mi> </mrow> <mi>τ</mi> <mo>∝</mo> <msup> <mrow> <mrow> <mi mathvariant="normal">V</mi> </mrow> </mrow> <mrow> <mrow> <mfrac> <mrow> <mrow> <mn>1</mn> <mo>+</mo> <mi>γ</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </mfrac> </mrow> </mrow> </msup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow></mrow> </mtd> <mtd> <mrow> <mfrac> <mrow> <mrow> <msub> <mrow> <mrow> <mi>τ</mi> </mrow> </mrow> <mrow> <mrow> <mi>f</mi> </mrow> </mrow> </msub> </mrow> </mrow> <mrow> <mrow> <msub> <mrow> <mrow> <mi>τ</mi> </mrow> </mrow> <mrow> <mrow> <mi mathvariant="normal">i</mi> </mrow> </mrow> </msub> </mrow> </mrow> </mfrac> <mo>=</mo> <msup> <mrow> <mrow> <mfenced separators="|"> <mrow> <mrow> <mfrac> <mrow> <mrow> <mn>2</mn> <mtext> </mtext> <mi mathvariant="normal">V</mi> </mrow> </mrow> <mrow> <mrow> <mtext> </mtext> <mi mathvariant="normal">V</mi> </mrow> </mrow> </mfrac> </mrow> </mrow> </mfenced> </mrow> </mrow> <mrow> <mrow> <mfrac> <mrow> <mrow> <mn>1</mn> <mo>+</mo> <mi>γ</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </mfrac> </mrow> </mrow> </msup> <mo></mo> <mrow> <mi mathvariant="normal"> </mi> </mrow> <mfrac> <mrow> <mrow> <msub> <mrow> <mrow> <mi>τ</mi> </mrow> </mrow> <mrow> <mrow> <mi mathvariant="normal">f</mi> </mrow> </mrow> </msub> </mrow> </mrow> <mrow> <mrow> <msub> <mrow> <mrow> <mi>τ</mi> </mrow> </mrow> <mrow> <mrow> <mi mathvariant="normal">i</mi> </mrow> </mrow> </msub> </mrow> </mrow> </mfrac> <mo>=</mo> <mo> (</mo> <mn>2</mn> <msup> <mrow> <mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mrow> <mfrac> <mrow> <mrow> <mn>1</mn> <mo>+</mo> <mi>γ</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </mfrac> </mrow> </mrow> </msup> </mrow> </mtd> </mtr> </mtable> </math> <!- [endif]-><!- [endif]->

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Under an adiabatic process, the volume of an ideal gas gets doubled. Consequently the mean collision time between the gas molecule changes from $τ_{1}$ to $τ_{2}$ . If $\frac{C_{p}}{C_{v}} = γ$ for the gas then a good estimate for $\frac{τ_{2}}{τ_{1}}$ is given by

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Under an adiabatic process, the volume of an ideal gas gets doubled. Consequently the mean collision time between the gas molecule changes from τ 1 to τ 2 . If C p C v = γ for the gas then a good estimate for τ 2 τ 1 is given by

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Under an adiabatic process, the volume of an ideal gas gets doubled. Consequently the mean collision time between the gas molecule changes from $τ_{1}$ to $τ_{2}$ . If $\frac{C_{p}}{C_{v}} = γ$ for the gas then a good estimate for $\frac{τ_{2}}{τ_{1}}$ is given by