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The following equation gives the relation between entropy, heat and temperature: $$ \frac{\Delta Q}{T} = \Delta S $$

So when you add a certain heat at a certain temperature, you increase the entropy of that system according to this equation if I understand correctly. If you rearrange this equation you get:

$$ \frac{\Delta Q}{\Delta S} = \frac{\partial Q}{\partial S} = T $$

Is this a meaningful equation or does it have some caveats? Also, what would it physically represent? Seems like it suggests that the temperature is determined by how much the heat changes (?) with respect to entropy gained. If you plot a $S,Q$ diagram where $S$ is the independent variable, the temperature is represented by the slope. However what kind of experiment would this be and what is macroscopically and microscopically happening?

bananenheld
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2 Answers2

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“$\Delta Q$” and “$dQ$” imply that there’s some function $Q$ that we can reasonably take the difference or derivative of. This isn’t the case; we can instead talk about $Q$, the heating, or $q$ (sometimes written $\delta Q$, for example), an infinitesimal amount of heating. So we have

$$\frac{q}{dS}=\frac{Q}{\Delta S}=T,$$

which are perfectly valid relations that apply to reversible heating at temperature T. Here, the heating expands the population of energy states, quantified by an increase in entropy. Yes, this does provide a way to measure the system temperature, if the other parameters are known and we attempt to experimentally approach the idealized case of reversible heating.

Chemomechanics
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One example is the change of phase, but with slowly heat input to have a reversible process. In typical situations there is also change in the volume and a constant pressure. So, using the first law:

$$dU = \delta q - \delta w = Tds - PdV \implies dU + PdV = TdS$$ $dU + PdV = \Delta H$, the enthalpy. $$T = \frac{\partial H}{\partial S}$$

The (constant) temperature of phase change corresponds to a relation of how much enthalpy changes for a given change of entropy. The relation is linear in this case.

It is also a way to measure the change of entropy along the process, by measuring the enthalpy variation. Because the relation is linear, we can use the total change of the process:$$\Delta S = \frac{\Delta H}{T}$$

Claudio Saspinski
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