I'm confused with some of the pressure and temperature implications of compressing gases with minute, incremental changes and doing it suddenly. Below is an exercise I've been attempting:
Here is my thought process:
In the case of (a), this seems to be an isothermal process. I'm assuming the gas is enclosed by a thermal reservoir of constant $T$, an the gas is being compressed incrementally, or reversibly, such that each small change $dV$ in volume there is a subsequent supplying of $dP$, in this case $dV$ is negative and $dP$ is positive. Since $T$ is constant, $P_1V_1 = P_2V_2$, so if $V_2 = 1/2V_1$, $P_2 = 2P_1$.
In this new case (b), the compression is done quickly, not at step $dV$ then $dP$, but several contributions of $dV$ before a contribution of $dP$. This means there is an imbalance, and since $PV \ne K$ where $K$ is a constant at every turn, $T$ must not be conserved. If $T$ is not conserved, there is a $\Delta U$ s.t. $ \Delta U = Q + W$. $W$ is positive and contributing here. Since $\Delta U \propto T \propto P$, then $P$ will increase.
However, I have the following issues with my logic:
- $V \propto T$ by the same equation I used for my deduction in the second bullet point: $PV=nk_bT$. And volume decreases. How am I then justified in saying this?
- How does this imply the pressure will be higher than (a)?
I'm seeking any advice on my issues, and whether anything in my thought process is wrong other than the ones pertaining to my issues.