"Under an adiabatic process, the volume of an ideal gas gets doubled. Consequently the mean collision time between the gas molecule changes from ${\tau _1}$ to ${\tau _2}$ . If ${{{C_p}} \over {{C_v}}} = \gamma$ for this gas then a good estimate for ${{{\tau _2}} \over {{\tau _1}}}$ is given by :"

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Accepted Answer

"$\tau$ $\propto$ ${V \over {\sqrt T }}$ ....(1)

Also we know, PV^$\gamma$ = k

We know, PV = nRT

$\Rightarrow$ P $\propto$ ${T \over V}$

$\therefore$ $\left( {{T \over V}} \right)$V^$\gamma$ = k

$\Rightarrow$TV^{$\gamma$ - 1} = k

$\Rightarrow$ T $\propto$ V^{1 - $\gamma$}

Using this value in equation (1)

$\tau$ $\propto$ ${V \over {{V^{{{1 - \gamma } \over 2}}}}}$

$\Rightarrow$ $\tau$ $\propto$ ${{V^{1 - {{1 - \gamma } \over 2}}}}$

$\Rightarrow$ $\tau$ $\propto$ ${{V^{{{\gamma + 1} \over 2}}}}$

$\therefore$ ${{{\tau _2}} \over {{\tau _1}}}$ = ${\left( {{{2V} \over V}} \right)^{^{{{\gamma + 1} \over 2}}}}$ = ${\left( 2 \right)^{{{1 + \gamma } \over 2}}}$"

Under an adiabatic process, the volume of an ideal gas gets doubled. Consequently the mean collision time between the gas molecule changes from ${\tau _1}$ to ${\tau _2}$ . If ${{{C_p}} \over {{C_v}}} = \gamma$ for this gas then a good estimate for ${{{\tau _2}} \over {{\tau _1}}}$ is given by :

Solution

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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 ${\tau _1}$ to ${\tau _2}$ . If ${{{C_p}} \over {{C_v}}} = \gamma$ for this gas then a good estimate for ${{{\tau _2}} \over {{\tau _1}}}$ is given by :

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