| Laws of thermodynamics |
|---|
| Zeroth Law |
| First Law |
| Second Law |
| Third Law |
| Fundamental Relation |
In thermodynamics, the fundamental thermodynamic relation expresses an infinitesimal change in internal energy in terms of infinitesimal changes in entropy, and volume for a closed system in thermal equilibrium in the following way.
Here, E is internal energy, T is absolute temperature, S is entropy, P is pressure, and V is volume. As all physics equations, this equation can be used in any unit system. In a consistent unit system like the SI system the corresponding equation for the numerical values of the physical quantities relative to the unit system is of the same form.
Derivation from the first and second laws of thermodynamics
The first law of thermodynamics states that:
According to the second law of thermodynamics we have for a reversible process:
Hence:
By substituting this into the first law, we have:
Letting dW be reversible pressure-volume work, we have:
This equation has been derived in the case of reversible changes. However, since E, S, and V are thermodynamic functions of state, the above relation holds also for non-reversible changes. If the system has more external variables than just the volume that can change and if the numbers of particles in the system can also change, the fundamental thermodynamic relation generalizes to:
Here the Xi are the generalized forces corresponding to the external variables xi. The μj are the chemical potentials corresponding to particles of type j.
Derivation from first principles
The above derivation uses the first and second laws of thermodynamics. The first law of thermodynamics is essentially a definition of heat, i.e. heat is the change in the internal energy of a system that is not caused by a change of the external parameters of the system.
However, the second law of thermodynamics is not a defining relation for the entropy. The fundamental definition of entropy of an isolated system containing an amount of energy of E is:
![S = k \log\left[\Omega\left(E\right)\right]\,](http://wpcontent.answers.com/math/c/0/4/c04285070a2bf9a8397a5d9e90b6076c.png)
where 
is the number of quantum states in a small interval between E and E + δE. Here δE is a macroscopically small energy interval that is kept fixed. Strictly speaking this means that the entropy depends on the choice of δE. However, in the thermodynamic limit (i.e. in the limit of infinitely large system size), the specific entropy (entropy per unit volume or per unit mass) does not depend on δE. The entropy is thus a measure of the uncertainty about exactly which quantum state the system is in, given that we know its energy to be in some interval of size δE.
Deriving the fundamental thermodynamic relation from first principles thus amounts to proving that the above definition of entropy implies that for reversible processes we have:
The fundamental assumption of statistical mechanics is that all the states are equally likely. This allows us to extract all the thermodynamical quantities of interest. The temperature is defined as:
![\frac{1}{k T}\equiv\beta\equiv\frac{d\log\left[\Omega\left(E\right)\right]}{dE}\,](http://wpcontent.answers.com/math/8/b/7/8b77282aaf38833c81c764e85b369215.png)
See here for the justification for this definition. Suppose that the system has some external parameter, x, that can be changed. In general, the energy eigenstates of the system will depend on x. According to the adiabatic theorem of quantum mechanics, in the limit of an infinitely slow change of the system's Hamiltonian, the system will stay in the same energy eigenstate and thus change its energy according to the change in energy of the energy eigenstate it is in.
The generalized force, X, corresponding to the external variable x is defined such that Xdx is the work performed by the system if x is increased by an amount dx. E.g., if x is the volume, then X is the pressure. The generalized force for a system known to be in energy eigenstate Er is given by:
Since the system can be in any energy eigenstate within an interval of δE, we define the generalized force for the system as the expectation value of the above expression:
To evaluate the average, we partition the energy eigenstates by counting how many of them have a value for 
within a range between Y and Y + δY. Calling this number 
, we have:
The average defining the generalized force can now be written:
We can relate this to the derivative of the entropy w.r.t. x at constant energy E as follows. Suppose we change x to x + dx. Then will change because the energy eigenstates depend on x, causing energy eigenstates to move into or out of the range between E and E + δE. Let's focus again on the energy eigenstates for which lies within the range between Y and Y + δY. Since these energy eigenstates increase in energy by Y dx, all such energy eigenstates that are in the interval ranging from E - Y dx to E move from below E to above E. There are
such energy eigenstates. If 
, all these energy eigenstates will move into the range between E and E + δE and contribute to an increase in Ω. The number of energy eigenstates that move from below E + δE to above E + δE is, of course, given by 
. The difference
is thus the net contribution to the increase in Ω. Note that if Y dx is larger than δE there will be the energy eigenstates that move from below E to above E + δE. They are counted in both 
and , therefore the above expression is also valid in that case.
Expressing the above expression as a derivative w.r.t. E and summing over Y yields the expression:
The logarithmic derivative of Ω w.r.t. x is thus given by:
The first term is intensive, i.e. it does not scale with system size. In contrast, the last term scales as the inverse system size and will thus vanishes in the thermodynamic limit. We have thus found that:
Combining this with
Gives:
which we can write as:
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