Entropy and heat

Reference: Daniel V. Schroeder, An Introduction to Thermal Physics, (Addison-Wesley, 2000) – Problem 3.28.

The thermodynamic identity for an infinitesimal process is

\displaystyle dU=TdS-PdV \ \ \ \ \ (1)

This relation bears a resemblance to heat plus work relation for the change in internal energy:

\displaystyle dU=Q+W \ \ \ \ \ (2)

In a quasistatic process, where a gas is being compressed slowly enough that the pressure has a chance to equalize throughout the volume of the gas at each stage, the work done in compressing the gas is {-PdV} (this is positive, as the volume decreases in compression so {dV<0}). In that case, then

\displaystyle W \displaystyle = \displaystyle -PdV\ \ \ \ \ (3)
\displaystyle Q \displaystyle = \displaystyle TdS \ \ \ \ \ (4)

and the change in entropy can be calculated as

\displaystyle dS=\frac{Q}{T} \ \ \ \ \ (5)

which agrees with the original definition of entropy.

As an example, suppose we have a litre of air at room temperature (300 K) and atmospheric pressure ({10^{5}\mbox{ N m}^{-2}}), and we heat it at constant pressure until it doubles in volume. From the ideal gas law, if {P} is constant and {V} doubles, then {T} must also double. The entropy change is therefore

\displaystyle \Delta S \displaystyle = \displaystyle C_{P}\int_{T_{i}}^{T_{f}}\frac{dT}{T}\ \ \ \ \ (6)
\displaystyle \displaystyle = \displaystyle C_{P}\ln\frac{T_{f}}{T_{i}}=C_{P}\ln2 \ \ \ \ \ (7)

where {C_{P}} is the heat capacity at constant pressure. From the appendix to Schroeder’s book, {C_{P}\approx29\mbox{ J K}^{-1}} for one mole of air (the values for nitrogen and oxygen are both around 29). The number of moles of air in one litre is

\displaystyle n \displaystyle = \displaystyle \frac{PV}{RT}\ \ \ \ \ (8)
\displaystyle \displaystyle = \displaystyle \frac{10^{5}\times10^{-3}}{\left(8.314\right)\left(300\right)}\ \ \ \ \ (9)
\displaystyle \displaystyle = \displaystyle 0.04\mbox{ mol} \ \ \ \ \ (10)

The change in entropy is therefore

\displaystyle \Delta S=\left(0.04\right)\left(29\right)\ln2=0.81\mbox{ J K}^{-1} \ \ \ \ \ (11)

One thought on “Entropy and heat

  1. Pingback: Entropy of diamond and graphite | Physics pages

Leave a Reply

Your email address will not be published. Required fields are marked *