Fundamental thermodynamic relation

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Laws of thermodynamics
Zeroth Law
First Law
Second Law
Third Law
Fundamental Relation
v • d • e

In thermodynamics, the fundamental thermodynamic relation is a mathematical summation of the first law of thermodynamics and the second law of thermodynamics subsumed into a single concise mathematical statement as shown below:

dE= T dS - P dV\,

Here, E is internal energy, T is temperature, S is entropy, P is pressure, and V is volume.

Thermodynamic derivation

Starting from the first law:

dE = \delta Q - \delta W\,

From the second law we have for a reversible process:

dS = \delta Q/T\,

Hence:

\delta Q = TdS\,

By substituting this into the first law, we have:

dE = TdS - \delta W\,


Letting dW be reversible pressure-volume work, we have:

dE = T dS - P dV\,

This 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:

dE = T dS - \sum_{i}X_{i}dx_{i} + \sum_{j}\mu_{j}dN_{j}\,

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 using the microcanonical ensemble

The above derivation can be criticized on the grounds that it merely defines a partitioning of the change in internal energy in heat and work in terms of the entropy. As long as we don't rigorously define the entropy in terms of the microscopic degrees of freedom of the system, the fundamental thermodynamic relation is vacuous.

The entropy of an isolated system containing an amount of energy of E is defined as:

S = k \log\left[\Omega\left(E\right)\right]\,
\Omega\left(E\right)\Omega\left(E\right)\frac{1}{k T}\equiv\beta\equiv\frac{d\log\left[\Omega\left(E\right)\right]}{dE}\,

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:

X = -\frac{dE_{r}}{dx}

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:

X = -\left\langle\frac{dE_{r}}{dx}\right\rangle\,
\Omega\left(E\right)
\Omega\left(E\right)=\sum_{Y}\Omega_{Y}\left(E\right)\,

The average defining the generalized force can now be written:

X = -\frac{1}{\Omega\left(E\right)}\sum_{Y} Y\Omega_{Y}\left(E\right)\,
\Omega\left(E\right)
N_{Y}\left(E\right)=\frac{\Omega_{Y}\left(E\right)}{\delta E} Y dx\,
Y dx\leq\delta E
N_{Y}\left(E\right) - N_{Y}\left(E+\delta E\right)\,
N_{Y}\left(E\right)

Expressing the above expression as a derivative w.r.t. E and summing over Y yields the expression:

\left(\frac{\partial\Omega}{\partial x}\right)_{E} = -\sum_{Y}Y\left(\frac{\partial\Omega_{Y}}{\partial E}\right)_{x}= \left(\frac{\partial\left(\Omega X\right)}{\partial E}\right)_{x}\,

The logarithmic derivative of Ω w.r.t. x is thus given by:

\left(\frac{\partial\log\left(\Omega\right)}{\partial x}\right)_{E} = \beta X +\left(\frac{\partial X}{\partial E}\right)_{x}\,

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:

\left(\frac{\partial S}{\partial x}\right)_{E} = \frac{X}{T}\,

Combining this with

\left(\frac{\partial S}{\partial E}\right)_{x} = \frac{1}{T}\,

Gives:

dS = \left(\frac{\partial S}{\partial E}\right)_{x}dE+\left(\frac{\partial S}{\partial x}\right)_{E}dx = \frac{dE}{T} + \frac{X}{T} dx\,

which we can write as:

dE = T dS - X dx\,