Foundation lesson
Entropy and temperature
Add a little energy to a body at fixed volume and its entropy rises by that energy divided by its absolute temperature. The same energy raises the entropy of a cold body more than that of a hot one.
Written for this edition, not translated from Einstein.
What does temperature have to do with entropy?
Add a small amount of heat to a body, slowly and reversibly, and keep its volume fixed so that it does no work. Its energy rises by that amount, and its entropy rises by the energy divided by its absolute temperature. So one joule raises the entropy of a body at 100 kelvin by 0.01 joule per kelvin, and of a body at 1000 kelvin by only 0.001: the same energy counts for more in a cold body. In symbols, with dE a small gain of energy and T the temperature:
At fixed volume and the stated constraints, entropy change is energy change divided by absolute temperature.
Read the other way round, 1/T is how steeply a body's entropy climbs as its energy grows.
The subscript V matters. If the volume can change, the body can do work while it takes in energy, and the relation gains another term. The derivative describes one kind of change, at fixed volume, not every process.
The relation also says why bodies in contact end at one temperature. Share a fixed total of energy between two bodies, and the total entropy is greatest when moving a little energy from one to the other no longer changes it: when their entropy slopes, and so their temperatures, are equal. §3 of the light-quanta paper uses the same relation for radiation, setting its entropy slope equal to 1/T.
Open the foundation: Partial derivatives and held-fixed quantities
Worked example: Three joules into a reservoir held at 300 kelvin
- A reservoir so large that its temperature stays at 300 kelvin receives 3 joules of heat, slowly.
- Its entropy rises by 3 J ÷ 300 K = 0.01 joule per kelvin.
- The same 3 joules given to a reservoir at 150 kelvin would raise its entropy by 0.02 joule per kelvin, twice as much.
- A small body warms as it takes in heat, so its temperature changes on the way. Its entropy change is found by adding dE/T over the warming, not by dividing by one temperature.
The relation fixes how entropy changes, not its absolute value: two entropy formulas that differ by a constant have the same slope everywhere, and a separate condition is needed to fix the constant. For an entropy per unit volume, that constant multiplied by two different volumes can change the difference between two states.
Try it: the entropy slope is one over the temperature
Wien's law gives the density ρ of light of one frequency ν at a temperature T: ρ = αν³e−βν/T. §3 of the light paper asks how fast the radiation's entropy density grows as its density grows, at fixed volume and frequency, and the answer has to be 1/T. Here is the check at one state, 600 terahertz and 3000 kelvin.
- Solved for the exponent, Wien's law says ln(ρ/αν³) = −βν/T.
- The entropy density of that light is −(ρ/βν)(ln(ρ/αν³) − 1). Its slope with respect to ρ is −(1/βν) ln(ρ/αν³), because the derivative of ρ ln ρ − ρ is ln ρ.
- Put in the exponent: the slope is (βν/T) ÷ βν = 1/T, whatever βν is. At 3000 K that is 1 ÷ 3000 = 3.333 × 10⁻⁴ per kelvin.
- At 600 THz, βν = hν/k is about 28 800 K, so βν/T is about 9.60: the light is dilute, well inside the range where Wien's law holds.
The radiation entropy workbench, LQ-04, starts at this state, with the volume unchanged. Its table shows the temperature recovered from the density: 3000 K at both ends. Open the radiation entropy workbench.
Show the workbench here
What it shows, in words
The temperature can be read from how an entropy changes: at fixed volume, the entropy of a body rises by the energy it gains divided by its temperature. For radiation obeying Wien's law the slope of the entropy density with respect to the energy density comes out exactly one over the temperature, at every frequency where the law holds. At 600 terahertz and 3000 kelvin it is 3.333 × 10⁻⁴ per kelvin.
Where this lesson stops
Before using 1/T as an entropy slope, say what is held fixed. Before comparing two entropies, say whether they are totals or per unit volume, and what fixes the constant.
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