- #1
f(x)
- 182
- 0
«Challenging» Thermodynamics Problem
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_0The 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}
dQ=dW+dU
dQ= \frac{i_{th}}{\varphi}
<br /> PV=nR \theta \mbox{ideal gas eqn}
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.
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_0The 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}
<br /> 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.
Last edited:
