Adiabatic Process

An adiabatic process is one in which absolutely no heat enters or leaves the system at any stage: $Q = 0$ throughout. This can come about in two rather different ways in practice: either the system is genuinely thermally insulated from its surroundings by a very good insulating (adiabatic) wall, so that heat simply has no path along which to flow; or the process happens so rapidly that there is not enough time for any appreciable heat exchange to take place, even though the system is not perfectly insulated -- the sudden compression of air in a bicycle pump, or the rapid expansion of gas escaping a punctured tyre, are both effectively adiabatic for this second reason.

Change in internal energy and temperature. Setting $Q = 0$ in the first law, $Q = \Delta U + W$, gives immediately

During an adiabatic expansion, the gas does positive work ($W>0$) on its surroundings, and since no heat is coming in from anywhere to replace the energy spent doing this work, that work must be paid for entirely out of the gas's own internal energy -- so $\Delta U$ is negative, and the gas's temperature falls. Conversely, during an adiabatic compression, work is done on the gas by the surroundings ($W<0$), and with no heat able to escape, all of this work goes directly into raising the gas's internal energy -- so $\Delta U$ is positive, and the gas's temperature rises. This is exactly why a bicycle pump grows noticeably warm as air inside it is compressed rapidly, and why gas escaping rapidly from a high-pressure cylinder feels cold as it expands.

Equation of an adiabatic process. Setting $dQ=0$ in the infinitesimal first law $dQ = C_v\,dT + P\,dV$ (using $dU=C_v\,dT$ for an ideal gas) gives $C_v\,dT = -P\,dV$; combining this with the differential form of the ideal gas equation of state, and using $\gamma = C_p/C_v$ together with the relation $C_p - C_v = R$ from Section 11.6, this differential equation can be integrated to give, for a reversible adiabatic process,

Using $PV = nRT$ to eliminate $P$ or $V$ in turn from this relation gives two further, equally useful, equivalent forms:

Which of these three forms is most convenient to use depends on which pair of quantities (out of $P$, $V$, $T$) a particular problem gives and asks for.

Work done. Starting from $dW = P\,dV$ and using $PV^{\gamma} = \text{constant}$ to write $P$ in terms of $V$, integrating between an initial state $(P_1,V_1,T_1)$ and a final state $(P_2,V_2,T_2)$ gives the work done by the gas in an adiabatic process:

(the second form follows from the first using $PV=nRT$ at each end state). Notice that this formula automatically has the correct sign: if the gas expands and cools, $T_1 > T_2$, so $W$ comes out positive (the gas does positive work), consistent with $\Delta U = -W$ being negative. …