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# Chemical equilibrium

Chemical equilibrium is a state where all chemical reactions proceed at the same rate as their reverse, so there is no change in the proportions of the various compounds. A common example given is the Haber-Bosch process, in which hydrogen and nitrogen combine to form ammonia. Equilibrium is reached when the rate of production of ammonia equals its rate of decomposition. Le Chatelier's principle describes qualitative predictions that can be made about chemical equilibrium.

Without energy input chemical reactions always proceed towards equilibrium. For a reaction

$kA + mB \leftrightarrow nC + pD$

equilibrium occurs when

$\frac{\left[A\right]^k \left[B\right]^m} {\left[C\right]^n \left[D\right]^p} = K$

where K is a constant called the equilibrium constant. The left side of the equation is called the mass action expression and is denoted Q for a generic state (not necessarily in equilibrium). For a single-step reaction, this can easily be derived just by considering the kinetics involved. Unlike rate equations, though, it still holds for multi-step reactions since the expressions for each step just multiply together. This, by the way, also gives us the relationship between equilibrium and temperature:

$K_T = K_\infty e^{-\frac{\Delta E}{RT}}$

where ΔE is the difference in energy per mole between reactants and products, e is the base of the natural logarithm, and R is the molar gas constant. The constant is mainly influenced by entropy change, but that's a little more complicated - whereas energy is roughly constant against concentration, entropy varies logarithmically so we have to refer back to a particular state. The relationship makes the most sense in terms of the free energy difference, ΔF* = ΔE - TΔS*, which represents the total work that can be done by the system as it develops. At equilibrium ΔF = 0, which gives us

$\Delta F^* = RT \ln {\left(\frac{Q^*}{K}\right)}$

Very often we consider the standard state, where Q = 1 in appropriate units, which can then be neglected. Note that all this applies to a reaction at constant temperature only. For a reaction at constant pressure (which is actually somewhat more typical) you would use the Gibbs free energy, ΔG* = ΔH - TΔS*, where ΔH is the change in enthalpy.

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