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• Physics

# Thermodynamic temperature

The thermodynamic temperature is the temperature defined using only the first and second laws of thermodynamics. It therefore does not depend on the thermometers used to measure the temperature of a body and its definition is thus universal and unambiguous. It coincides with the absolute temperature measured using perfect gas thermometers.

To understand its origin, consider a reversible heat engine executing a Carnot cycle, exchanging a quantity Q1 and Q2 of heat with two sources at different temperatures t1 and t2.

It does not matter what thermometer is used at this stage provided it is always the same one.

The second law requires that, in this situation, the heat exchanges for all the possible reversible engines satisfy the following universal relationship which depends only on the temperatures t1 and t2.

$\frac{Q_1}{Q_2} = f(t_1,t_2)$

It is easy to show that we must have the relationship:

$\frac{f(t_0,t_2)}{f(t_0,t_1)} = f(t_1,t_2)$

For if we apply the first formula, we will have the two equations:

$\frac{Q_1}{Q_0} = f(t_1,t_0)$

$\frac{Q_2}{Q_0} = f(t_2,t_0)$

Which, on taking the quotient, gives:

$\frac{f(t_0,t_2)}{f(t_0,t_1)} = \frac{Q_2}{Q_1}$

Consider t0 as a fixed reference temperature. If a suitable constant K is introduced to normalise the conversion of heat units and temperature units, a new temperature scale can be introduced such that:

$Kf(t_0,t)=\theta (t)$

Which makes it possible to write

$\frac{Q_1}{Q_2} = f(t_1,t_2) = \frac{\theta (t_2)}{\theta (t_1)}$

There is now nothing to prevent from using the above universal function theta as the definition of the temperature: this is the thermodynamic temperature:

$T=\theta (t)$

Finally, the well known thermodynamic relationships for an arbitrary reversible heat engine running according to a Carnot cycle are obtained:

$\frac{Q_1}{Q_2} = f(t_1,t_2) = \frac{T_1}{T_2}$

$\frac{Q_1}{T_1} = \frac{Q_2}{T_2}$

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