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# Activity Coefficient

For a reaction to take place there should be an activity. Energy is required for this activity. The activity coefficient is the most important and fundamental property in the thermodynamic study of liquid mixtures. It is a measure of the deviation of the behavior of a component in a mixture from ideal behavior, and it has been interpreted by various theories of liquid mixtures.

Most solutions are not ideal solutions. This non-ideality may arise from a difference in either entropy of mixing or enthalpy of mixing. With water being a solvent any hydrophobic solute would result in a highly non-ideal mixture owing to the difference between the water solute interactions and the sum of the solute-solute and water-water enthalpic interactions.

The addition of hydrogen bonding substituents to the organic compound would result in a lesser deviation from the ideal behavior. The extent of deviation from the ideal behavior is expressed by the activity coefficient.

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## Standard State Conditions

Either it is available in the system itself or it has to be supplied by the surroundings, referred to as the standard state conditions. Certain reactions are instantaneous or spontaneous and will stop only when the reaction is complete. In other words, all the reactants get converted into products, if they are in molecular proportions. Certain reactions take place slowly and they will end in equilibrium.

The activity of the substances involved can be measured in these later reactions. Vapour pressure is one of the bases for all measurements of chemical equilibria in solutions. In a solution, there is a relationship between the vapour pressure of the solute and the vapour pressure of the solvent. Henry’s law and Raoult’s law describe these relationships, which form the basis of all measurements of chemical equilibria in solutions.

## Henry's Law

This law was originally proposed to explain the solubility of gases in water. It states that "the solubility is directly proportional to the pressure of the gas".

The proportionality is used to define the Henry’s law constant KH = partial pressure of gas/mole fraction in solution. Larger the KH value, lower will be the solubility at a fixed partial pressure of the gas.

### Standard state and activity

• A reference state of a substance to which all the other states can be related is called its standard state. For gases we can take 1 atmosphere as the standard state since most gases nearly follow the ideal gas law at this pressure.
• In a mixture of gases, their individual partial pressures can be used as concentration units to relate each gas to its standard state.
• The solutions of non electrolytes like benzene and toluene are considered in their standard states. The mole fraction of each of them in their pure state is a convenient unit to measure it in solution.
• In the case of ionic substances like NaCl in water, pure water can be taken in its standard state. Sodium chloride behaves differently in the solid state and in the dissolved state. In a solution or dissolved state, the Na+ and Cl- ions (solute particles ) are surrounded by water molecules and these ions are different from the Na+ and Cl- ions in NaCl (s).
• For a standard state there should not be any solute-solute interactions. In case of NaCl in solution there is solute – solute interactions and a standard state can be achieved only if the concentration is less than 10-4. This dilution is very high for any reference.
• Standard state is attained when the solution becomes ideal.
• In order to arrive at the standard state in such dilutions, a new standard state is created based on Henry’s law for dilute solutions.
• This new concentration unit is called ACTIVITY.
For dilute solutions of ionic nature the activity is given as a ratio between pressure of solute and molality of solute instead of mole fraction.

KH = pressure of solute/molality of solute = Psolute/molality …………….(A)

Pressure of solute is the vapour pressure of Na+ and Cl- ions above the solution. For the standard state of electrolyte solutions, an extension up to 1 molal is drawn and a hypothetical standard state is created. This is termed as hypothetical standard state because at 1 m concentration solute - solute interaction will be there.

The vapour pressure of Na+ and Cl- ions above this standard state is

P standard state = P0 solute = KH X I molal

A Direct measurement of vapour pressures of Na+ and Cl- ions above NaCl is not possible. This is required to find the standard state. Only when the standard state of the solute is known, we can define the activity of the solute. This results in another theoretical equation.

Activity of the solute = P solute / P standard state
= P solute / KH X I molal

The equilibrium constant expressions are properly defined using activities as units of concentration.

As long as equation (A) holds good

Activity of dilute solute = KH X molality / KH X I molal
= molality/ 1 molal

This shows that the ACTIVITY of dilute solution is same as molality except for the dimension associated for molality is removed. Thus as a unit of concentration, activity is only approximately related to molality if the solution is concentrated, while it is equal to molality if the solution is dilute.

Although important quantities such as activity and standard state depend upon vapour pressure which cannot be measured directly, these depend on thermodynamic equations.

## Raoult's Law

Henry’s law applies to the solute in solution, activity of the solute and the standard state of solutes. Raoult’s law deals with the solvent in a solution. He proposed that the vapour pressure of the solvent is reduced by the presence of solute and the reduction in vapour pressure is proportional to the mole fraction of the solvent.

P solvent $\propto$ X solvent
where, X is the mole fraction

P solvent = X solvent . P0 solvent

where, P0 is the vapour pressure of pure solvent

This relationship shows that the vapour pressure of the solution depends upon the number of moles of solute present in a solution. Such property is called colligative property. By measuring the colligative property, the molecular weight of the solute can be determined.

## Ionic Activity

It is necessary to count the calculations based upon the activity and activity coefficient and not on concentration. In the measurement of physiological parameters like the serum electrolyte calculations the activities and activity coefficients are required. In clinical medicines more accurate measurements of body constituents can be obtained by taking into consideration the activity and activity coefficient. Dealing with ions the activity is termed as IONIC ACTIVITY and certain ion specific electrodes will give most accurate ionic activity.

Ionic concentration is measured in milli equivalents per litre. This is a measure of number of ions in one litre of solution. Ionic activity reflects also the interaction with its environment. Ionic activity ‘a’ is thus a product of activity coefficient ($\gamma$ I ) and the concentration (c).

a = $\gamma$ I c

In qualitative terms ionic activity can be termed as effective concentration and activity coefficient as degree of deviation from ideal behavior. Ionic activity can be expressed in two scales- molar and molal. Molar scale is used in clinical analysis since it is easy to make solutions and dilute them. Molal scale is generally for water based solutions.

Ionic activity can be measured by ion selective electrodes. These electrodes select a particular ion in a multi ionic solution.