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;

Ask & Answer Question

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 -

$\frac{1}{2}$

2 Views | Posted 2 months ago

Asked by Shiksha User

1 Answer

Answered by

alok kumar singh | Contributor-Level 10

2 months 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]->

0 Upvotes 0 Downvotes

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

1 Answer

Similar Questions for you

Learn more about...

Most viewed information

Share Your College Life Experience

Didn't find the answer you were looking for?

Need guidance on career and education? Ask our experts

Your Question

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

1 Answer

Similar Questions for you

Learn more about...

Most viewed information

Share Your College Life Experience

Didn't find the answer you were looking for?

Need guidance on career and education? Ask our experts

Your Question

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