Activity and Fugacity

Fugacity is defined as thermodynamic pressure, which is used to find the properties of real component by deviation from its ideal gas properties. For an ideal gas f = P and fi = yiP for ideal gas solution.

Activity is relative fugacity, which is the ratio of fugacity to fugacity in standard state. 

The property of real solution can be calculated by its deviation either from ideal gas or ideal solution behavior. the deviation from ideal gas is taken in terms of fugacity while the deviation from ideal solution is taken in terms of activity.

Fugacity is generally used to find the deviation in gaseous phase while activity is used to find deviation in liquid and solid.
The gas phase or vapor phase property is calculated by using fugacity coefficient while the liquid phase property is calculated by using activity coefficient.

Thermodynamics Properties of solution

The thermodynamic properties of solution can be estimated from either ideal gas behavior or ideal solution. The deviation corresponds to ideal gas is termed as residual properties while deviation from ideal solution is termed as excess properties.
the solution to behave as ideal gas should follow the relation for any component:
partial molar volume = molar volume of pure = RT/P
while for ideal solution: partial molar volume = molar volume of pure
the condition for ideal solution is less restrictive than ideal gas behavior i.e components of similar physical properties or present in excess form ideal solution.

The fugacity for ideal solution can be calculated using Lewis Randall rule when the component is in excess. but for other component which has low composition follows Henry's law. and in between the real solution properties should be calculated using fugacity coefficient and activity coefficient.
as shown in figure, solution behaves according to Henry's Law when xi <= 0.1, and follows Lewis Randall  rule when xi >= 0.9.

Partial molar property

The molar property of a component changes from its pure state property, when it is solution. for example the molar volume of pure water is 18.079 cm3/mol and when one mole of water is added to large volume of ethanol, the volume changes by 14 cm3, i.e. partial molar volume of water at infinite dilution is 14 cm3. the difference occurs due to the interaction between the particles. when water is pure, its molecules are surrounded by same kind of molecules, but in a water ethanol solution, water molecule is surrounded by both water molecules and ethanol molecules. The interaction forces in solution are different, thereby molecules have different partial properties. The interaction depends on the quantity of like and dislike molecules. So, partial molar property is composition dependent.
the ideal solution is defined as which has same interaction as in pure component. Therefore the molecules which have similar molecular properties, form an ideal solution.