A-level Chemistry/AQA/Module 5/Thermodynamics/Thermodynamic Temperature

What is Temperature?
There are a few ways we can describe temperature. But a large amount of them are surprisingly poor. Sadi Carnot for instance imagined heat as a liquid, and it was this view that allowed him to develop heat engine theory.

Here we will develop a way of viewing temperature that makes it useful in a Thermodynamic point of View.

Two systems in Thermal Contact
Imagine two systems, 1 and 2, with total energy E = E1 + E2 and Entropy S = S1 + S2.

Conservation of energy of the system means that E2 = E - E1 always.

Differentiating by E1 we see that:

$$ \begin{matrix} \frac{d E_2}{d E_1} &=& \frac{d(E - E_1)}{dE_1} \\ &=& -\frac{d E_1}{dE_1} \\ &=& -1 \end{matrix} $$

Now if we look at Entropy changing with Energy. $$\frac{\partial S}{\partial E_1} = \frac{d S_1}{d E_1} + \frac{d S_2}{d E_1} = \frac{d S_1}{d E_1} + \frac{d S_2}{d E_2}\frac{d E_2}{d E_1} = \frac{d S_1}{d E_1} - \frac{d S_2}{d E_2}$$

At equilibrium there is no change in entropy, so when the systems are at the same temperature $$\frac{\partial S}{\partial E_1} = \frac{d S_1}{d E_1} - \frac{d S_2}{d E_2} = 0 $$

We use this to define Temperature. $$ T^{-1} = \frac{\partial S}{\partial E}_{N,V}$$ N and V means the Volume and Number of Particles is fixed.

Or for a specific system $$ T_i^{-1} = \frac{\partial S_i}{\partial E_i}_{N_i,V_i}$$

Usage
If we were to look at how entropy changed with temperature, it is easy to show that as entropy always increases, energy always flows from the hotter body to the cooler one. Try it.

Authors
--Frontier 11:51, 23 Apr 2005 (UTC)

Note
One of the equations wasn't parsing correctly at the time of writing. I've left it as is because the error message was 'Unknown Function \begin' which should be recognised. I'm not really sure what the problem is.