# Challenging Thermodynamics Problem

«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 $\varphi$ $\mbox{At time} \ t=0$, it contains air at atmospheric pressure $p_a$ and temperature $\theta_0$The temperature of the surrounding is $\theta_a ( > \theta_0 )$ Find the amount of gas in moles in the vessel at time t. Take $C_v = \frac{5R}{2}$

## Homework Equations

$$dQ=dW+dU$$
$$dQ= \frac{i_{th}}{\varphi}$$
$$PV=nR \theta \mbox{ideal gas eqn}$$

## The Attempt at a Solution

I assumed pressure to be constant throughout the problem.
$$P=P_a$$
Initially,
$$i_{th} = \frac{\theta_a - \theta_0}{\varphi}$$
Now since volume and pressure both are constant,
PV=const.
or,
$nRd\theta + R\theta dN = 0$

$\frac{d\theta}{\theta} = -\frac{dn}{n}$

Now i try to apply first law, which gives,
$$\frac{\theta_a - \theta}{\varphi} dt = nC_vd\theta + \theta C_v n$$ where $\theta\ is\ temperature \ at\ time \ t$
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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what level is this question?

what level is this question?
Senior secondary

Any1 got an idea?

Long time....if any1 knows a tactic for this do tell

Mabye you should try posting this in the Advanced Physics forum instead?