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 ncert solutions class 11th Physics Class 11th

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 -

$\frac{1}{2}$

3 Views | Posted 2 months ago

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1 Answer

Answered by

alok kumar singh | Contributor-Level 10

2 months ago

Correct Option - 3

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