Assume there is a function of systems at equilibrium, which gives it's heat energy. So if X is a certain amount of substance in a certain equilibrium state, we have: E(X) as the total heat energy.

Obviously E(nX) = nE(X) where n is a multiplier -- or the total

Now the defining property of heat flow is it always flows from hot to cold. So if there is no heat flow, the systems must have the same temperature. This is the Zeroth Law.

Even though E(2X) > E(X), both 2X and X must be at the same temperature. T(nX) = T(X)

Now lets take two different items at different temperatures and put them in contact. E(X) and E(Y).
After a time, heat flow trickles to a stop and X reaches a new equilibrium state at a new temperature. X->X'. Likewise, Y->Y'. And the temperatures are the same. T(X') = T(Y'). But why that particular temperature?

For that, we need the Second Law of Thermodynamics, the one about entropy. Like energy, the amount of entropy of a substance in a particular state is proportional to the amount of the substance. So S(nX) = nS(X)

So T(X') = T(Y') because if significant amounts of energy would to flow from X' to Y' (or vice versa) the total entropy would decrease. So at this equilibrium temperature T(X') = T(Y') = t, only infinitesimal amounts of of energy can be exchanged, with no change of entropy. So a tiny change in the energy of X', ∂E(X') can flow to Y' and cause the oppose tiny change in engery, -∂E(Y'), only if the change in entropy is also equal and opposite, ∂S(X') = -∂S(Y').

So at a certain temperature, t, equal and opposite tiny changes to E produce equal and opposite changes in S.

So it follows, at this particular temperature, t, that ∂S(X')/∂E(X') = ∂S(Y')/∂E(Y'), and since this has to be true for any two substances, then ∂S(X')/∂E(X') = f(t). From just the zeroth law and the second law, we can't say what this function of temperature is f(t) is, but it has to be the same for all substances.

But we have no obvious meters that read S. And no meters, therefore that read ∂S or f(t). So it is up to physicists to make the world not only predictable, but simply predictable. So we will define the definitions in a way that makes the math simplest as possible.

By definition, ∂S(X)/∂E(X) = 1/T(X) -- and you can see this is hinted by the units. S is measured in Joules per Kelvin, and E in Joules.

Combining this with the First Law, we get ∂E(X) = T(X)∂S(X) - Work, which is just a a restatement of conservation of energy. For a gas, work might be Pressure × ∂Volume.

Hi

Thank ou very much for your reply but I'm afraid it was way over my head.

Since I posted this i found a really good simple explanation which i have posted below

Temperature is a number that is related to the average kinetic energy of the molecules of a substance. If temperature is measured in Kelvin degrees, then this number is directly proportional to the average kinetic energy of the molecules.

Heat is a measurement of the total energy in a substance. That total energy is made up of not only of the kinetic energies of the molecules of the substance, but total energy is also made up of the potential energies of the molecules.

When heat, (i. e., energy), goes into a substance one of two things can happen:

1. The substance can experience a raise in temperature. That is, the heat can be used to speed up the molecules of the substance. Since Kelvin temperature is directly proportional to the average kinetic energy of molecules in a substance, an factor increase in temperature causes an equal factor increase in the average kinetic energy of the molecules. And if the kinetic energy of the molecules increase, the speed of the molecules will increase, although these increases are not directly proportional. The kinetic energy of a body is proportional to the square of the speed of the body.

2. The substance can change state. For example, if the substance is ice, it can melt into water. Perhaps surprisingly, this change does not cause a raise in temperature. The moment before melting the average kinetic energy of the ice molecules is the same as the average kinetic energy of the water molecules a moment after melting. Although heat is absorbed by this change of state, the absorbed energy is not used to speed up the molecules. The energy is used to change the bonding between the molecules. Changing the manner in which the molecules bond to one another constitutes a change in potential energy. Heat comes in and there is an increase in the potential energy of the molecules. Their kinetic energy remains unchanged.

So, when heat comes into a substance, energy comes into a substance. That energy can be used to increase the kinetic energy of the molecules, which would cause an increase in temperature. Or that heat could be used to increase the potential energy of the molecules causing a change in state that is not accompanied by an increase in temperature.

Right, so heat is the total stored kinetic and potential energy of the substance which is not proportional to temperature, but does go up when the temperature does. (But with phase transitions, sometimes the heat energy changes even though the temperature does not.)