﻿

# Fundamental thermodynamic 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. $dU= T dS - P dV\,$

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

## Derivation from the first and second laws of thermodynamics

The first law of thermodynamics states that: $dU = \delta Q - \delta W\,$

According to the second law of thermodynamics we have for a reversible process: $dS = \delta Q/T\,$

Hence: $\delta Q = TdS\,$

By substituting this into the first law, we have: $dU = TdS - \delta W\,$

Letting dW be reversible pressure-volume work, we have: $dU = T dS - P dV\,$

This equation has been derived in the case of reversible changes. However, since U, S, and V are thermodynamic functions of state, the above relation holds also for non-reversible changes. If the system has more external parameters 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: $dU = T dS - \sum_{i}X_{i}dx_{i} + \sum_{j}\mu_{j}dN_{j}\,$

Here the Xi are the generalized forces corresponding to the external parameters xi. The μj are the chemical potentials corresponding to particles of type j.

## Derivation from statistical mechanical 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: $S = k \log\left[\Omega\left(E\right)\right]\,$

where $\Omega\left(E\right)$ 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: $dS =\frac{\delta Q}{T}$

The fundamental assumption of statistical mechanics is that all the $\Omega\left(E\right)$ states are equally likely. This allows us to extract all the thermodynamical quantities of interest. The temperature is defined as: $\frac{1}{k T}\equiv\beta\equiv\frac{d\log\left[\Omega\left(E\right)\right]}{dE}\,$

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. 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 parameter 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: $X = -\left\langle\frac{dE_{r}}{dx}\right\rangle\,$

To evaluate the average, we partition the $\Omega\left(E\right)$ energy eigenstates by counting how many of them have a value for $\frac{dE_{r}}{dx}$ within a range between Y and Y + δY. Calling this number $\Omega_{Y}\left(E\right)$, we have: $\Omega\left(E\right)=\sum_{Y}\Omega_{Y}\left(E\right)\,$

The average defining the generalized force can now be written: $X = -\frac{1}{\Omega\left(E\right)}\sum_{Y} Y\Omega_{Y}\left(E\right)\,$

We can relate this to the derivative of the entropy with respect to x at constant energy E as follows. Suppose we change x to x + dx. Then $\Omega\left(E\right)$ 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 $\frac{dE_{r}}{dx}$ 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 $N_{Y}\left(E\right)=\frac{\Omega_{Y}\left(E\right)}{\delta E} Y dx\,$

such energy eigenstates. If $Y dx\leq\delta E$, 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 $N_{Y}\left(E+\delta E\right)$. The difference $N_{Y}\left(E\right) - N_{Y}\left(E+\delta E\right)\,$

is thus the net contribution to the increase in Ω. Note that if Y dx is larger than δE there will be energy eigenstates that move from below E to above E + δE. They are counted in both $N_{Y}\left(E\right)$ and $N_{Y}\left(E+\delta E\right)$, therefore the above expression is also valid in that case.

Expressing the above expression as a derivative with respect to 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 with respect to 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 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: $dE = T dS - X dx\,$

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Thermodynamic system — A thermodynamic system is a precisely defined macroscopic region of the universe, often called a physical system, that is studied using the principles of thermodynamics. All space in the universe outside the thermodynamic system is known as the… …   Wikipedia

• Thermodynamic potential — A thermodynamic potential is a scalar potential function used to represent the thermodynamic state of a system. One main thermodynamic potential which has a physical interpretation is the internal energy, U. It is the energy of configuration of a …   Wikipedia

• Thermodynamic beta — In statistical mechanics, the thermodynamic beta is a numerical quantity related to the thermodynamic temperature of a system. The thermodynamic beta can be viewed as a connection between the statistical interpretation of a physical system and… …   Wikipedia

• Thermodynamic cycle — Thermodynamics …   Wikipedia

• Entropy — This article is about entropy in thermodynamics. For entropy in information theory, see Entropy (information theory). For a comparison of entropy in information theory with entropy in thermodynamics, see Entropy in thermodynamics and information… …   Wikipedia

• Heat capacity — Thermodynamics …   Wikipedia

• Helmholtz free energy — In thermodynamics, the Helmholtz free energy is a thermodynamic potential which measures the “useful” work obtainable from a closed thermodynamic system at a constant temperature and volume. For such a system, the negative of the difference in… …   Wikipedia

• Enthalpy — Thermodynamics …   Wikipedia

• Laws of thermodynamics — The laws of thermodynamics, in principle, describe the specifics for the transport of heat and work in thermodynamic processes. Since their conception, however, these laws have become some of the most important in all of physics and other… …   Wikipedia

• First law of thermodynamics — In thermodynamics, the first law of thermodynamics is an expression of the more universal physical law of the conservation of energy. Succinctly, the first law of thermodynamics states: DescriptionThe first law of thermodynamics basically states… …   Wikipedia