An ideal gas undergoes a process in which PV-a - constant- where V is the volume occupied by the gas initially at pressure P- At the end of the process- rms speed of gas molecules has become a1-2 times of its initial value- The value of CV so that energy transferred by the heat to the gas becomes -a- times of the initial energy isR1-aa-1-naa2-1- The value of n is

PV^ a = constant Molar heat capacity in the process nbsp;is given by C = C_V + R/1 x (for PV^x = constant) nbsp; = C_V + R/1 ( a) = C_V + R/1 + a At the end ...

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Standard XII

Physics

RMS and KE Expressions

An ideal gas ...

Question

An ideal gas undergoes a process in which $P V^{- a}$ = constant, where $V$ is the volume occupied by the gas initially at pressure $P$ . At the end of the process, rms speed of gas molecules has become $a^{\frac{1}{2}}$ times of its initial value. The value of $C_{V}$ so that energy transferred by the heat to the gas becomes ' $a$ ' times of the initial energy is
$R (\frac{(1 + a)}{a - 1} - \frac{n a}{a^{2} - 1}]$ . The value of $n$ is

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Solution

$P V^{- a} =$ constant
Molar heat capacity in the process is given by
$C = C_{V} + \frac{R}{1 - x}$ (for $P V^{x} =$ constant)
$= C_{V} + \frac{R}{1 - (- a)}$
$= C_{V} + \frac{R}{1 + a}$
At the end of the process $V_{r m s}$ is $a^{\frac{1}{2}}$ times initial value
$\Rightarrow$ temperature is 'a' times intial value $(∵ V_{r m s} \propto T^{\frac{1}{2}})$
$Δ Q = n C Δ T = n C T (a - 1)$
$= n T (a - 1) [C_{V} + \frac{R}{1 + a}]$
But it is given that $Δ Q = a P V$
$\Rightarrow a P V - \frac{n R T (a - 1)}{a + 1} = n (a - 1) C_{V} T$
$\Rightarrow a P V - \frac{(a - 1)}{a + 1} P V = n (a - 1) C_{V} T$
$(∵ P V = n R T)$
$\Rightarrow (a^{2} + 1) P V = n (a^{2} - 1) C_{V} T$
$\Rightarrow C_{V} = \frac{P V (a^{2} + 1)}{n T (a^{2} - 1)} = \frac{R (a^{2} + 1)}{(a^{2} - 1)}$

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