# Activities and molar concentrations

I would be really happy if someone would clarify the term "chemical activity" and its implications for me, and also how it is related to the concept of molar concentration. My textbook fails at explaining this clearly and, sadly, the Internet is filled with vague definitions and scientific verbiage that's too difficult for me to grasp. Also, why are activities included in various formulae, and what are their uses?

I never quite actually understood what are activities and how do they differ from molar concentrations, nor how are they used. In addition, when working with the Nernst equation, why are we allowed to compute the activities of the reactants and the equilibrium concentrations of the products? This is the stuff that really baffled me and I'd be highly appreciative if you could clarify it.

Activity (a) is a pure mathematical construct used in thermodynamics and molarity (M) is a practical concept which can conveniently carry out in the lab. Someone can tell you to prepare 0.02 M Na$$^{+1}$$ solution but not 0.02 a Na$$^{+1}$$ solution. You should rather ask, how activity and concentration related? If you open a physical chemistry text, the definition provided there is abstract and good for regurgitating in an exam only.

Before activity can be explained in a less abstract way, one has to introduce another concept called fugacity. I will quoted original wording to the inventor- G. N. Lewis the person who gave the concept of activity to the world. It is always good to see the original work to see what was going on in the mind of the inventor.

Quote from his paper:

The idea of fugacity is thus evolved from the use of vapor pressure as a measure of escaping tendency. When a substance is in equilibrium with its vapor, the fugacity, in order to fulfil the laws of escaping tendency, must be the same i both. The fugacity of a substance is therefore equal to its vapor pressure if the vapor behaves like a perfect gas. Speaking in terms not very precise, we may say that the fugacity of a substance is equal to the vapor pressure that the substance $$x$$ could have if its vapor were a perfect gas. It has been shown in the preceding paper that for a given substance in a given state the fugacity is a definite property of which the numerical value can in most cases be readily determined, and which is well suited to serve as an exact measure of the escaping tendency.

The unit of fugacity is pressure as you should have guessed by now. Now come to activity. Lewis wanted to express an equivalent concept for solutions so that its unit was same concentration. He defined activity as:

$$a =\frac{fugacity}{RT}$$

with the following properties:

Besides these we shall use the following definitions of the activity, as per Lewis:

1. When the activity of a substance is the same in two phases, that substance will not of itself pass from one phase to the other.

2. When the activity of a substance is greater in one phase than in another, the substance will pass from the one phase to the other, when they are brought together.

3. The activity of a perfect gas is equal to its concentration. The activity of the solute in a perfect solution, at constant temperature and pressure, is proportional to its concentration.

4. We shall see that these statements suffice to define the activity of a substance in any state, and except in unusual cases enable us to calculate its numerical value.

Most thermodynamic equations are derived assuming a perfect gas or ideal solution. However, for real a solution the chemical potential $$\mu_i=\mu^0_i +RT\ln(x_i)$$ may not be accurate for the solvent where $$\mu^0_i$$ is the chemical potential of the liquid solvent at 1 atm and $$x_i$$ is the mole fraction. The solute may, similarly, not follow ideal behaviour. The activity is introduced to allow us to use the familiar equations but now as $$\mu=\mu^0_{liq} +RT\ln(\gamma x_i)$$ where $$\gamma$$ is the activity coefficient thus $$\mu_i=\mu^0_{liq} +RT\ln(a_i)$$ where $$a=\gamma x$$. Thus we can think of activity as effective concentration or effective pressure (now called fugasity) relative to its standard state. The activity coefficient is unity in the ideal case. The activity has to be measured experimentally because $$\gamma$$ depends on exactly what intermolecular interactions exits as any temperature and pressure and cannot be easily calculated. As an example, naphthalene dissolved in benzene behaves almost ideally and the activity coefficient $$\gamma = 1$$ but when dissolved in hexane this is not the case and $$\gamma=2.5$$. A positive deviation from Raoult's law indicates that the vapour pressure is larger than that expected from perfect behaviour and so the activity is larger than the mole fraction and $$\gamma >1$$.

Activity is a concept created to make theory match the measurements. For example, the pH is a concept that can be measured with a glass electrode or a Hydrogen electrode. At high dilution, the measured tension of the cell, or the measured pH is exactly described by the formula pH = - log ]H+]. At higher concentration, the measured tension, or the measured pH is not exactly -log[H+]. Some chemists say that the measured pH is the log of a sort of "apparent" H+ concentration, that is called "activity". The activity may be considered as a sort of concentration calculated by dividing the number of mole of H+ by the volume of the free water, and not by the total volume of the solution. Free water is the water which is not attached around the ions of the solute by electrostatic forces. The free water occupies a smaller volume than the whole solution. This volume is nearly impossible to determine, so that we prefer speaking of activity, being $$\ce{10^{-pH}}$$, and that is all.