# The Gibbs Energy

Entropy is the fundamental basis for assigning the direction of spontaneous change, but to use it we must consider the entropy changes of both the system and the surroundings, which is somewhat inconvenient.

It is possible to devise a method which relates the entropy change of the surroundings to properties of the system, thus allowing us to focus solely on the properties of the system while still taking the change in the surroundings into account.

We start with a system at the same temperature, T, as its surroundings and in thermal equilibrium with them, and consider what happens when there is a transfer of heat from the system to the surroundings. From the Second Law, we know that  dStot  ³  0 for a spontaneous process, and thus we may write dSsys + dSsur ³ 0 . This implies:

We have already formulated the expression dSsur  =  (dqsur) / T . Logically, dqsur  =  -dqsys , as the heat transferred to the system must come from the surroundings, or vice versa. Substitution of this into the expression for dSsur  gives   dSsur  =  -(dqsys) / T. This gives us the requirement for a spontaneous change as:

which is the Clausius inequality. This rearranges very simply to

This expresses the criterion for spontaneous change purely in terms of state functions of the system.

We will now consider the heat transfer as occurring at constant pressure, and make the assumption that there is no work other than expansion work. This allows us to write dqp  =  dH , which, upon substitution into the Clausius inequality gives:

This inequality may be rewritten to consider systems where either the enthalpy or entropy are constant:

The interpretation of these equations remains in line with the Second Law. At constant pressure, for a system where the enthalpy remains constant, there is no exchange of heat with the surroundings (dSsur = 0). Thus for a spontaneous change, the entropy of the system must not decrease if the Second Law is to be obeyed. This is confirmed by the above equation.

Likewise, if the entropy of the system does not change, the entropy of the surroundings must not decrease if the the Second Law is to be obeyed. This means there can be no transfer of heat from the surroundings to the system, i.e. the enthalpy of the system must not increase.

The inequality relating dSsys and dHsys may be rewritten as    dHsys  – T dSsys  £  0 . This may be expressed more simply by the introduction of the thermodynamic quantity called the Gibbs Energy, G. It is defined as

When the state of the system changes at a constant temperature, this gives us

which in turn means that the criterion of spontaneous change at constant pressure and temperature is given by: