Internal energy

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For a body or system with well-defined boundaries, the internal energy, denoted by U, or sometimes E, is the total of the kinetic energy due to the motion of molecules (translational, rotational, vibrational) and the potential energy associated with the vibrational and electric energy of atoms within molecules or crystals. It includes the energy in all the chemical bonds, and the energy of the free, conduction electrons in metals. One can also calculate the internal energy of electromagnetic or blackbody radiation. It is a state function of a system, an extensive quantity. The SI unit of energy is the joule although other historical, conventional units are in still use, such as the (small and large) calorie for heat.

Internal energy does not include the translational or rotational kinetic energy of a body as a whole. It also does not include the relativistic mass-energy equivalent E = mc², which would drop out in considering changes in any event. It excludes any potential energy a body may have because of its location in an external gravitational or electrostatic field, although the potential energy it has in a field due to an induced electric or magnetic dipole moment does count, as does the energy of deformation of solids (stress-strain).

The principle of equipartition of energy in classical statistical mechanics states that each molecular degree of freedom receives 1/2 kT of energy, a result which was modified when quantum mechanics explained certain anomalies; e.g., in the observed specific heats of crystals (when hν > kT). For monatomic helium and other noble gases, the internal energy consists only of the translational kinetic energy of the individual atoms. Monatomic particles, of course, do not (sensibly) rotate or vibrate, and are not electronically excited to higher energies except at very high temperatures.

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

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Heat and work

The absolute value of the internal energy U cannot be measured. There is an arbitrary zero reference value, just as there is for gravitational potential energy; the internal energy is defined only up to an arbitrary additive constant. Only its change can be measured, and this is

<math> \Delta U = Q + W \, </math>

where

Q is heat added to a system (measured in joules in SI); that is, a positive value for Q represents heat flow into a system while a negative value denotes heat flow out of a system.

W is the work done on a system (measured in joules in SI)

ΔU is the value of the internal energy after a process minus its value before. Among physicists and chemists there has been disagreement about the sign convention for the (±) W term; but in either case, a positive value for its term, including the sign, means work done on a system. The historical ambiguity arises from the fact that the most common type of work considered is −PΔV where P is the (external) pressure and V is the volume of the system. Work is being done on a system if it contracts and its volume decreases (ΔV < 0). The minus sign reverses this, making the second term properly positive in the expression for ΔU. Aside from the anomaly of pressure-volume work, it is clearer if we use +W. So we have

<math> \Delta U = Q -\ P\Delta V + W' \, </math>

where W' represents all the kinds of work other than −PΔV. There are many such kinds of work that one may want to consider in thermodynamics: electrical (electrochemical cells), magnetic, force × distance, stress × strain (requiring tensors), etc., which allow us to include structured, non-bulk systems. The expression for ΔU is absolutely general and applies to all systems which are closed to matter exchange with their surroundings.

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Entropy

There is always a change in the entropy of a system directly related to any gain or loss of heat

<math> Q = T (\Delta S)_{heat} \,</math>

where T is the absolute temperature of the system. Heat (conduction, radiation, etc.) is the only way entropy can be exchanged with the surroundings for a system closed to matter exchange. But the entropy can also change due to internal irreversible processes. The second law of thermodynamics says that all spontaneous processes create entropy, so this contribution is inherently positive; or in the limit of "reversible" processes or at equilibrium, zero

<math> (\Delta S)_{internal} \ge 0 \,.</math>

Since the total entropy change is

<math> \Delta S= (\Delta S)_{heat} +(\Delta S)_{internal} \,</math>

we arrive at the inequality

<math> Q \le T\Delta S \, .</math>

The discrepancy is what Clausius called the "uncompensated heat". Consequently

<math> \Delta U \le T\Delta S + W \,. </math>

At constant T this can be rearranged to give an expression in terms of the Helmholtz free energy A = U - TS

<math> \Delta U - T\Delta S = (\Delta A)_{T} \le W \, </math>

which is the potential for the maximum amount of isothermal work of all kinds that can be obtained from a system

<math> -W \le -(\Delta A)_{T} \, .</math>

For infinitesimal changes, all the equations above can be written with Δ replaced by d, indicating differential forms.

[edit]

Internal energy

From Free net encyclopedia

Heat and work

Entropy

See also

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