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**«Challenging» Thermodynamics Problem**

## Homework Statement

Consider a cubical vessel of edge a, having a small hole in one of its walls. The total thermal resistance of the wall is [itex] \varphi [/itex] [itex] \mbox{At time} \ t=0 [/itex], it contains air at atmospheric pressure [itex] p_a [/itex] and temperature [itex] \theta_0 [/itex]The temperature of the surrounding is [itex] \theta_a ( > \theta_0 ) [/itex] Find the amount of gas in moles in the vessel at time t. Take [itex] C_v = \frac{5R}{2} [/itex]

## Homework Equations

[tex] dQ=dW+dU [/tex]

[tex]dQ= \frac{i_{th}}{\varphi} [/tex]

[tex]

PV=nR \theta \mbox{ideal gas eqn} [/tex]

## The Attempt at a Solution

I assumed pressure to be constant throughout the problem.

[tex] P=P_a [/tex]

Initially,

[tex] i_{th} = \frac{\theta_a - \theta_0}{\varphi} [/tex]

Now since volume and pressure both are constant,

PV=const.

or,

[itex]nRd\theta + R\theta dN = 0[/itex]

[itex] \frac{d\theta}{\theta} = -\frac{dn}{n} [/itex]

Now i try to apply first law, which gives,

[tex] \frac{\theta_a - \theta}{\varphi} dt = nC_vd\theta + \theta C_v n [/tex] where [itex] \theta\ is\ temperature \ at\ time \ t[/itex]

But since these rates are also varying, i have no idea how to continue. Specially if someone could throw light on the integration part.

Thanks for any assistance.

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