# Which pressure to use in the $T~ds$ equation?

Let's say I have an adiabatic, rigid, open container that has an amount of air at some pressure, $P_\text{cv}$, and some temperature, $T_\text{cv}$. I have heated pressurized air coming into the container. I am trying to calculate the change in specific entropy of the air in the container. I use the $T~ds$ equation, but I am not sure which pressure to use. The equation used is

$$T\frac{ds_\text{cv}}{dt}=\frac{du_\text{cv}}{dt}+P\frac{dv}{dt}.$$

Would the pressure term in the equation be the pressure of the incoming air, or would it be the pressure of the air in the control volume?

• You're pumping air into your container. You either need an outlet in your system (and a way to describe the heat carried away convectively), or you're going to need to have a $\mu dN$ term if this is going to have any chance of being right. – Jerry Schirmer Sep 8 '13 at 21:08
• We have never learned about $\mu dN$ in class and there is no outlet in the specified problem – Greg Harrington Sep 9 '13 at 4:15
• you've learned about $PV = NkT$, right? Well, you're changing $N$ by pumping gas, in, so $P,V$ and $T$ are going to have to change in response. – Jerry Schirmer Sep 9 '13 at 12:14