Challenging Thermodynamics Problem

1. Feb 23, 2007

f(x)

«Challenging» Thermodynamics Problem

1. The problem statement, all variables and given/known data

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 $C_v = \frac{5R}{2}$

2. Relevant equations

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

3. 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.

Last edited: Feb 24, 2007
2. Feb 23, 2007

eaboujaoudeh

what level is this question?

3. Feb 24, 2007

f(x)

Senior secondary

4. Feb 25, 2007

f(x)

Any1 got an idea?

5. Mar 31, 2007

f(x)

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

6. Mar 31, 2007

AznBoi

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