Isothermal vs. adiabatic compression of gas in terms of required energy

Question

I am working on an instruction manual of sorts to be used with an introductory course in thermodynamics. As an example of problem solving, I attempt to answer the following question:

If you have little energy available, would you rather use an isothermal or an adiabatic process to compress a gas?

My analysis is as follows:

The work required to compress a gas from volume $V_0$ to volume $V$ is $$W=-\int_{V_0}^{V}PdV$$ For the isothermal compression, the ideal gas law, $P=\frac{nRT}{V}$, is used and inserted into the equation above: $$W=-\int_{V_0}^{V}\frac{nRT}{V}dV=-nRT\int_{V_0}^{V}\frac{dV}{V}=nRT\ln\left(\frac{V_0}{V}\right)$$ For the adiabatic compression, $PV^\gamma=P_0V_0^\gamma \iff P=P_0\left(\frac{V_0}{V}\right)^\gamma$ is valid. Thus, the required work is $$W=-\int_{V_0}^{V}P_0\left(\frac{V_0}{V}\right)^\gamma dV=-\frac{P_0V_0^\gamma}{-\gamma+1}\left(V^{-\gamma+1}-V_0^{-\gamma+1}\right)=\frac{PV-P_0V_0}{\gamma-1}$$ $$W=-\frac{P_0V_0-PV}{\gamma-1}$$

This result indicates that the magnitude of the required work depends on the properties of the gas in question. Thus, it may vary which of the two compression processes requires the lowest amount of energy.

Now, finally, to my question:

Intuitively I would assume that the isothermal work is usually lower than the adiabatic work, as compression lowers the volume of the system and therefore usually increases the temperature if heat exchange with the surroundings is not allowed. Isothermal compression requires heat transfer to the surroundings to maintain constant temperature, lowering the pressure of the system and thus lowering the resistance to compression compared to the adiabatic compression (where heat exchange is not allowed).

Is the isothermal work actually smaller than the adiabatic one in most cases or is my argument flawed?

Answer 1

To solve this, try to use what I call the "graphical apparatus". For an isothermal process: $$ \begin{align} PV&=\text{constant}\\ P\mathrm{d}V&=-V\mathrm{d}P\\ \frac{\mathrm{d}P}{\mathrm{d}V}&=-\frac{P}{V}\\ \end{align}$$ for adiabatic process: $$ \begin{align} PV^\gamma&=\text{constant}\\ \frac{\mathrm{d}P}{\mathrm{d}V}&=-\gamma\frac PV \end{align}$$ Therefore, starting at the same point on a P-V graph, the curves for an adiabatic and isothermal processes will diverge and the adiabatic curve will have a steeper slope. For the same reduction in volume (the graph in the picture is for expansion, not for contraction. In case of contraction, the curves will be reversed, i.e. adiabatic curve will be above the isothermal curve, and will enclose greater area under it for the same reduction in pressure), more area will be enclosed by adiabatic, and since the area $\int P\mathrm{d}V$ gives the work required, isothermal work is smaller than adiabatic for the same reduction in volume.

Your argument is correct. To provide more mathematical support to it, you can observe the fact that it is both increase in temperature and reduction in volume which increases the pressure in adiabatic process and only reduction in volume increases pressure in isothermal process. The exponent of volume in adiabatic equation ($PV^\gamma=K$) is $\gamma>1$, as compared to 1 in the isothermal equation. Hence, pressure change is more sensitive to volume change and is larger in magnitude.

Answer 2

Your final equation for the isothermal work should read (you made a sign error):

$$W=nRT\ln{\frac{V_0}{V}}$$

You can make the comparison with the adiabatic case much more easily if you re-express the adiabatic work as: $$W=\frac{P_0V_0}{\gamma - 1}\left[\left(\frac{V_0}{V}\right)^{\gamma -1}-1\right]=\frac{nRT_0}{\gamma - 1}\left[\left(\frac{V_0}{V}\right)^{\gamma -1}-1\right]$$ But, for the isothermal case, $T=T_0$. So for the isothermal case, $$W=nRT_0\ln{\frac{V_0}{V}}$$ Now you can see how much more easily it would be to compare the two amounts of work.