# Fugacity and Activity

The free energy equation change ÎG = RTlog_{e} P_{2}/P_{1} is applicable only to ideal gases. When this equation applied to real gases, particularly at higher pressures, this expression does not reproduce the change in free energy because, under these conditions V is not equal to nRT/P. So, in order to apply this equation to non-ideal systems, Lewis introduced two new thermodynamic quantities, fugacity and activity.

Consider a system composed of liquid water and its vapour. Liquid water has a tendency to escape into the vapour phase while the vapour tends to escape the gaseous state and come into the liquid phase by condensation. When the system is in equilibrium, these two escaping tendencies become equal and we observe a constant vapour pressure at a constant temperature. In general, it may be stated that each substance in a given state has a tendency to escape from that state and this escaping tendency is called **fugacity** and is denoted by '*f* '. It is related to the free energy content (G) by the
expression

G = RT ln *f* + B

where B is a constant depending upon the temperature and the nature of the substance. It is not possible to evaluate B since the absolute values of the free energy are not known. To circumvent this difficulty, all free energy measurements for any given substance are referred to as standard reference point.

If we represent by GÂș the free energy per mole and *f* Âș the fugacity in this standard state, then GÂș is given by

G – GÂș = (RT ln *f* + B) − (RT ln *f* Âș + B)

or, G – GÂș = (RT ln *f* / *f* Âș

or, G = GÂș + RT ln *f* / *f* Âș ---Equation:1

The ratio *f* / *f* Âș is called **activity** and is denoted by the symbol 'a'. The activity of any substance may therefore, be defined as the ratio of fugacity of the substance in the given state to the fugacity of the same substance in the standard state.

G = GÂș + RT ln a

In the standard state,

G = GÂș

∴ RT ln a = 0

or, a = 1

i.e., *in the standard state the activity of a substance is equal to unity*.

In any other state, the value of activity will depend upon the difference (G – GÂș). The difference in free energy per mole caused on passing from one state in which the free energy is G_{1} and the activity a_{1} to another state in which these are G_{2} and a_{2} respectively, is given by the expression

ÎG = G_{2} – G_{1} = (GÂș + RT ln a_{2}) – (GÂș + RT ln a_{1})

or, ÎG = RT ln a_{2} / a_{1}

The similarity between the above equation and equation ÎG = RT 1n P_{2} / P_{1} suggests that **activity is the thermodynamic counterpart of the gas pressure**.

*For the standard state of any gas at the given temperature, the fugacity is taken as equal to unity*, viz, *f* Âș = 1 and on the basis of this definition the activity of any gas becomes equal to fugacity

a = *f*/*f* Âș = *f*/1 = *f*

So, now the equation:1 can be written as

G = GÂș + RT log_{e} *f*

*For an ideal gas, the fugacity is equal to pressure and f / P = 1. For a real gas, the fugacity is not equal to P and the ratio f / P varies*.

It has been observed that on decreasing the pressure, the behaviour of the gas approaches that of an ideal gas. Therefore, it may be stated that *f* approaches P as P approaches zero.

The ratio *f* / P is called fugacity coefficient of a gas and is represented by the symbol 𝛾. Fugacity coefficient is defined as the ratio of fugacity of a real gas to the actual pressure P of the gas. It is represented as-

*f*/P = đŸ

It is the measure of deviations of a real gas from ideal gas behavior. The units of fugacity and pressure are same therefore fugacity coefficient will be a pure number.

At extremely low pressure (when P approaches to zero) fugacity coefficient approaches to unity. All the gases approach ideality under this pressure range.

## Physical Significance of Fugacity

Considering the system in which liquid is in contact with the vapor phase. In this water molecules in the liquid escape into the vapor phase by evaporation while molecules in the vapor phase have the tendency to escape into liquid phase by condensation. These two escaping tendencies become equal at equilibrium. Thus it is accepted that any substance has the tendency to escape from its state. Lewis termed this escaping tendency as Fugacity.

How to Determine the Fugacity of gas?

Source: Essentials of Physical Chemistry by B.S.Bahl