3.2: Steps to improve efficiency: isothermal and adiabatic changes

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There are two special cases we need to consider in order to extend the way we construct these cycles beyond the constant volume (or isochoric) and the constant-pressure (or isobaric) circumstances.

One case concerns a process in which there’s no temperature change, but where work is done. We’ve said before that in most processes where there’s no temperature change involve no heat transfer - but we’ve also said that temperature most closely connects to the state of internal energy. When those two statements come into conflict, the state function “wins out” - in an isothermal process where work is done, there will be heat transfer; internal energy change is forced to zero.

To put it another way, if the temperature of a system (and therefore that system’s internal energy) is held constant, any heat you introduce to the system has to immediately be released from the system as work, and any work you do on the system has to immediately released from that system as heat.

We have, though, already found what work equals in a state of constant temperature:

$w = n R T \ln \frac{V_{i}}{V_{f}}$

If ΔU = 0, and q = -w, then this must also be true (note the swapping of subscripts):

$q = n R T \ln \frac{V_{f}}{V_{i}}$

This expression is referred to as the isothermal transfer of heat. Even though the pressure is changing, we only need the change in volume to complete the computation. And what’s more, it aids in generating efficiency; it is a step that encourages total transfer of heat energy to work performed.

We've also studied another way to improve efficiency: preventing heat losses at all. Recall that processes that proceed in a fashion that prevent heat loss are known as adiabatic processes. These are either processes that are instantaneous, so that work done happens so rapidly that heat doesn’t have time to transfer, or processes that are well-insulated, so temperature changes inside the container are unable to impact temperature outside of the container. In these cases, because q = 0, all work done on or by the system translates into the internal energy change of the system; ΔU = w.

We have been through the realization that adiabatic processes don’t neatly translate to holding a quantity constant in the same way that constant-volume, constant-pressure, or isothermal processes do. We have to intentionally derive the quantity that's being held constant.

It turns out that we've already laid the foundation for that derivation, when we worked out how volume changes with changing temperature in an adiabatic process:

$\frac{T_{f}}{T_{i}} = {(\frac{V_{i}}{V_{f}})}^{γ - 1} w h e r e γ = \frac{\bar{C_{P}}}{\bar{C_{V}}} (adiabatic process)$

We can rearrange this to group final and initial terms together on their respective sides of the equation:

$T_{f} V_{f}^{γ - 1} = T_{i} V_{i}^{γ - 1} (adiabatic process)$

$T V^{γ - 1} = c o n s t a n t (adiabatic process)$

It turns out (and this is left as an exercise to the reader) that we can derive a very similar equation by working out how pressure changes with changing volume in an adiabatic environment. Here, we recall Boyle's law where temperature is held constant in a closed container:

$upper P upper V equals c o n s t a n t left-parenthesis closed container at constant temperature right-parenthesis$

It turns out that, similar to Boyle's law, pressure does increase when the volume of the container decreases in an adiabatic process. However, in an adiabatic process, temperature is not intentionally held constant, and therefore the relationship looks subtly different, and reintroduces γ:

$upper P Subscript f Baseline upper V Subscript f Superscript gamma Baseline equals upper P Subscript i Baseline upper V Subscript i Superscript gamma Baseline left-parenthesis adiabatic process right-parenthesis$

$upper P upper V Superscript gamma Baseline equals c o n s t a n t left-parenthesis adiabatic process right-parenthesis$

The ratio γ is easily shown to be a precise ratio for ideal gases; it’s 5/3 for monatomic ideal gases, and 7/5 for diatomic. (Prove this!) For real gases, the ratio isn’t quite as neat, but it is viable to work with.

Being able to equate internal energy change with work done for no heat transfer makes the computation of work done very straightforward even if pressure isn’t held constant ( $Δ U = w = n \bar{C_{V}} Δ T$ ). And this gives us the tools to develop a heat engine that is much more efficient than the constant-pressure/constant-volume cycle that we described before.

Here is a table that summarizes the types of processes we have revealed so far, and that we can use as reference for determining internal energy changes, heat transfers, and work done:

Name of process	Constant quantity	$Δ U =$	$q =$	$w =$
Isobaric	$P$	$n \bar{C_{V}} Δ T$	$n \bar{C_{P}} Δ T$	$- P Δ V$
Isochoric	$V$	$n \bar{C_{V}} Δ T$	$n \bar{C_{V}} Δ T$	$0$
Isothermal	$T$	$0$	$n upper R upper T ln left-parenthesis upper V Subscript f Baseline slash upper V Subscript i Baseline right-parenthesis$	$n upper R upper T ln left-parenthesis upper V Subscript i Baseline slash upper V Subscript f Baseline right-parenthesis$
Adiabatic	$upper T upper V Superscript gamma minus 1 Baseline semicolon upper P upper V Superscript gamma$	$n \bar{C_{V}} Δ T$	$0$	$n \bar{C_{V}} Δ T$