Classical Thermodynamics in a Nutshell

Derivation of classical thermodynamics is usually difficult to understand, partly because of the way thermodynamics itself evolved. Thermodynamics was discovered in a romantic age (as opposed, for example, to the enlightenment period when Newton’s laws evolved), and the exposition consists of stories instead of mathematics. We have stories about refrigerators, furnaces, laws, forbidden processes, the universe, perpetual motion, and so forth. We are not so bold to pontificate this way today. 

In this study, we are considering what is called phenomenological variables. We are ignoring any statistical concepts completely, as if we were the original thermodynamic researchers. To start, we propose that bodies have internal thermal energy U, and that changes in U result from changes in volume V and empirical temperature θ (note: T is reserved for a special temperature):

This being true, U is at least a function of V and θ:

Volume is something we can measure explicitly, to measure temperature we need a device, a thermometer. In fact, we don’t experience temperature, we experience “hotness”, for example, boiling water is hotter than ice water. So, temperature is a measure of “hotness”, and any thermodynamic theory we develop must be consistent with this measure. We will assume we have a thermometer that measures an empirical temperature. As this theory unfolds, the correct type of temperature measurement will become apparent.

Another aspect of equation (2) is that different materials have different functions U. So, equation (1) is not a statement of thermodynamics, it is just a statement of calculus, and:

The difference between equation (1) and the “first law”, is that we identify two different mechanisms for changing the internal energy: working (dW) and heating (dQ):

For a fluid body, the mechanical working is given explicitly by pressure p times dV. 

Combining equation (1) with equations (4) and (5):

The heating is defined by the properties of the internal energy, pressure, and the definition of working. This is completely backwards from the way thermodynamics was actually developed. Equation (6) was called calorimetry, with f₁ called the heat capacity and f₂ + p called the latent heat (not to be confused with present usage of this term).

Note that equation (6) is called an inexact differential. What this really means is:

Or equivalently:

Before the concepts of energy were developed, heat was assumed to be conserved. Carnot, in his analysis of thermodynamics, explicitly assumed that heat was conserved, but that work came from the fall of heat from a high temperature to a lower temperature, as water falls from a height to drive a water wheel. Conservation of heat implies that dQ is, in fact, an exact differential, and equation (5) says pressure p does not depend on temperature, not a reasonable result.

The finding that Q was not a heat function did not diminish the desire for a heat function. There is a fixup for converting an inexact differential to an exact differential, called an integrating factor. If we can find a function g(V,θ), so that:

then g is an integrating factor if:

The new function S provides the heat function we desired, and:

It would be desirable for g to be a universal function for every fluid, but at this stage it is not apparent that g is special. If we suppose that g is universal, what might it be?

Let us choose a special case, which is known to be approximated by some dry pure gases:

This gas is called an ideal gas, and T is the ideal gas temperature. In this case, R, Cv, and U₀ are constants, which implies f₂ is zero. Using equations (12) in equation (10), we find:

We can solve equation (13) by separation of variables, namely g=H(V)M(T):

Since all the V dependent variables are on the left side and all the T dependent variables are on the right side, and given a constant c:

The first differential equation (15) depends on Cv, which is material specific, so this equation cannot be universal. The only way around this problem is to make c equal 0. The solution to the second differential equation (15) is M equals K/T, where K is a constant:

This solution implies that H is a constant equals H₀. The final result is:

Since g is a universal function, we can choose as a standard H₀K equal 1

Note that Fahrenheit temperature is scaled by the constant 5/9 to get Kelvin temperature, so the constant factor in equation (17) should not be problematic as long as we get our units right.   This gives the familiar result, the “second law” of thermodynamics, with the heat function S named the entropy:

For the special case of equation (12):