Fundamental thermodynamic relationFrom Wikipedia, the free encyclopedia

In thermodynamics, the fundamental thermodynamic relation is generally expressed as aninfinitesimal change in internal energy in terms of infinitesimal changes in entropy, and volume for aclosed 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.

This is only one expression of the fundamental thermodynamic relation. It may be expressed in otherways, using different variables (e.g. using thermodynamic potentials). For example, the fundamentalrelation may be expressed in terms of the Helmholtz free energy (F) as:

and in terms of the enthalpy as

Derivation from the first and second laws of thermodynamics

The first law of thermodynamics states that:

where and are infinitesimal amounts of heat supplied to the system by its surroundings and

work done by the system on its surroundings, respectively.

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:

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This equation has been derived in the case of reversible changes. However, since , , and arethermodynamic functions of state, the above relation holds also for non-reversible changes. If thesystem has more external parameters than just the volume that can change and if the numbers ofparticles in the system can also change, the fundamental thermodynamic relation generalizes to:

Here the are the generalized forces corresponding to the external parameters . The are thechemical potentials corresponding to particles of type .

Derivation from statistical mechanical principles

The above derivation uses the first and second laws of thermodynamics. The first law ofthermodynamics is essentially a definition of heat, i.e. heat is the change in the internal energy of asystem 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. Thefundamental definition of entropy of an isolated system containing an amount of energy of is:

where is the number of quantum states in a small interval between and . Here

is a macroscopically small energy interval that is kept fixed. Strictly speaking this means that theentropy depends on the choice of . However, in the thermodynamic limit (i.e. in the limit ofinfinitely large system size), the specific entropy (entropy per unit volume or per unit mass) does notdepend on . The entropy is thus a measure of the uncertainty about exactly which quantum statethe 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 thatthe 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:

This definition can be derived from the microcanonical ensemble, which is a system of a constantnumber 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, theenergy eigenstates of the system will depend on x. According to the adiabatic theorem of quantummechanics, in the limit of an infinitely slow change of the system's Hamiltonian, the system will stay inthe same energy eigenstate and thus change its energy according to the change in energy of theenergy eigenstate it is in.

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The generalized force, X, corresponding to the external parameter x is defined such that is thework performed by the system if x is increased by an amount dx. E.g., if x is the volume, then X is thepressure. The generalized force for a system known to be in energy eigenstate is given by:

Since the system can be in any energy eigenstate within an interval of , we define the generalizedforce 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 and . 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 with respect to 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 and . Let's focus

again on the energy eigenstates for which lies within the range between and . Since

these energy eigenstates increase in energy by Y dx, all such energy eigenstates that are in theinterval 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 rangebetween and and contribute to an increase in . The number of energy eigenstates thatmove from below to above is, of course, given by . The difference

is thus the net contribution to the increase in . Note that if Y dx is larger than there will beenergy eigenstates that move from below to above . They are counted in both

and , 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 theexpression:

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The logarithmic derivative of 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 asthe inverse system size and thus vanishes in the thermodynamic limit. We have thus found that:

Combining this with

Gives:

which we can write as:

External links

The Fundamental Thermodynamic Relation (http://theory.physics.manchester.ac.uk/~judith

/stat_therm/node38.html)

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Categories: Thermodynamics Statistical mechanics Thermodynamic equations

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