Reversible expansion

A reversible change in thermodynamics is a change that can be reversed by an infinitesimal modification of a variable. The key word ‘infinitesimal’ sharpens the everyday meaning of the word ‘reversible’ as something that can change direction. We say that a system is in equilibrium with its surroundings if an infinitesimal change in the conditions in opposite directions results in opposite changes in its state. One example of reversibility that we have encountered already is the thermal equilibrium of two systems with the same temperature. The transfer of energy as heat between the two is reversible because, if the temperature of either system is lowered infinitesim ally, then energy flows into the system with the lower temperature. If the temperature of either system at thermal equilibrium is raised infinitesimally, then energy flows out of the hotter system. Suppose a gas is confined by a piston and that the external pressure, pex, is set equal to the pressure, p, of the confined gas. Such a system is in mechanical equilibrium with its surroundings (as illustrated in Section 1.1) because an infinitesimal change in the external pressure in either direction causes changes in volume in opposite directions. If the external pressure is reduced infinitesimally, then the gas expands slightly. If the external pressure is increased infinitesimally, then the gas contracts slightly. In either case the change is reversible in the thermodynamic sense. If, on the other hand, the external pressure differs measurably from the internal pressure, then changing p_ex infinitesimally will not decrease it below the pressure of the gas, so will not change the direction of the process. Such a system is not in mechanical equilibrium with its surroundings and the expansion is thermodynamically irreversible. To achieve reversible expansion we set p_ex equal to p at each stage of the expansion. In practice, this equalization could be achieved by gradually removing weights from the piston so that the downward force due to the weights always matched the changing upward force due to the pressure of the gas. When we set p_ex = p, eqn 2.5 becomes

dw=−p_exdV=−pdV

(Equations valid only for reversible processes are labelled with a subscript rev.) Although the pressure inside the system appears in this expression for the work, it does so only because pex has been set equal to p to ensure reversibility. The total work of reversible expansion is therefore

We can evaluate the integral once we know how the pressure of the confined gas depends on its volume. Equation 2.10 is the link with the material covered in Chapter 1 for, if we know the equation of state of the gas, then we can express p in terms of Vand evaluate the integral.

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Poly (phenylene oxide) s

Higher-order structures

The convergence of energy levels

Absorption intensities

Butadiene and π-electron binding energy

Free expansion

The hydrogen cycle

Ring and cluster compounds

Resolutions can be carried out by chromatography on chiral materials

Axes and centres of symmetry

المرافق الإنزيمي حامض الليبويك Lipoic acid

التأثير التحريضي (الحاث) Inductive effect