# Internal energy

The internal energy of a system (abbreviated E or U) is the total kinetic energy due to the motion of molecules (translational, rotational, vibrational) and the total potential energy associated with the vibrational and electric energy of atoms within molecules or crystals. Internal energy is a quantifiable state function of a system. The SI unit of internal energy is the same as for energy, and is the joule.

For systems consisting of molecules, the internal energy is partitioned among all of these types of motion. In systems consisting of monatomic particles, such as helium gas and other noble gases, the internal energy consists only of the translational kinetic energy of the individual atoms. Monatomic particles, of course, do not rotate or vibrate, and are not excited to higher electrical energies, except at very high temperatures.

From the standpoint of statistical mechanics, the internal energy is shown to be equivalent to the ensemble average of the total energy of the system.

## Measurement

Internal energy can not be measured directly; it is only measured as a change (ΔU). The equation for change in internal energy is

[itex] \Delta U = Q \pm W \,\! [itex]

where

Q is heat input to or output of the system, measured in joules
W is work done on or by the system, measured in joules

A positive value of W represents work done on the system, a negative value representing work done by the system. Generally physisists use negative "U" indicating work done by a system; chemists use positive "U" indicating work done on a system. A positive value of Q represents heat flow into the system, a negative value representing heat flow out of the system.

## Duhem-Gibbs Relation

For quasistatic processes, the following relation holds:

[itex]dU = TdS - pdV + \mu dN \,\! [itex]

where

T is the temperature, measured in kelvins
S the entropy, measured in joules / kelvin
p the pressure, measured in pascals
V the volume, messured in cubic metres.
μ the chemical potential, (measured in joules or joules per mole)
N the number of particles in the system, (unitless or measured in moles)

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