Thus, they are essentially equations of state, and using the fundamental equations, experimental data can be used to determine sought-after quantities like G (Gibbs free energy) or H (enthalpy).
[1] The relation is generally expressed as a microscopic change in internal energy in terms of microscopic changes in entropy, and volume for a closed system in thermal equilibrium in the following way.
Here, U is internal energy, T is absolute temperature, S is entropy, P is pressure, and V is volume.
It may be expressed in other ways, using different variables (e.g. using thermodynamic potentials).
However, since U, S, and V are thermodynamic state functions that depend on only the initial and final states of a thermodynamic process, the above relation holds also for non-reversible changes.
of the chemical components, in a system of uniform temperature and pressure can also change, e.g. due to a chemical reaction, the fundamental thermodynamic relation generalizes to: The
If the system has more external parameters than just the volume that can change, the fundamental thermodynamic relation generalizes to Here the
Other generalized forces tend to increase their conjugate displacements.)
The fundamental thermodynamic relation and statistical mechanical principles can be derived from one another.
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
is the number of quantum states in a small interval between
is a macroscopically small energy interval that is kept fixed.
Strictly speaking this means that the entropy depends on the choice of
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
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
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
The temperature is defined as: This definition can be derived from the microcanonical ensemble, which is a system of a constant number of particles, a constant volume and that does not exchange energy with its environment.
Suppose that the system has some external parameter, x, that can be changed.
The generalized force, X, corresponding to the external parameter x is defined such that
is the work performed by the system if x is increased by an amount dx.
, we define the generalized force for the system as the expectation value of the above expression: To evaluate the average, we partition the
, we have: The average defining the generalized force can now be written: We can relate this to the derivative of the entropy with respect to x at constant energy E as follows.
with respect to 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 thus vanishes in the thermodynamic limit.
We have thus found that: Combining this with Gives: which we can write as: It has been shown that the fundamental thermodynamic relation together with the following three postulates[2] is sufficient to build the theory of statistical mechanics without the equal a priori probability postulate.
For example, in order to derive the Boltzmann distribution, we assume the probability density of microstate i satisfies
The normalization factor (partition function) is therefore The entropy is therefore given by If we change the temperature T by dT while keeping the volume of the system constant, the change of entropy satisfies where Considering that we have From the fundamental thermodynamic relation, we have Since we kept V constant when perturbing T, we have
Combining the equations above, we have Physics laws should be universal, i.e., the above equation must hold for arbitrary systems, and the only way for this to happen is That is It has been shown that the third postulate in the above formalism can be replaced by the following:[3] However, the mathematical derivation will be much more complicated.