Scientists have a mathematical way of measuring randomness - it is called entropy and is related to the number of arrangements of particles (such as molecules) that lead to a particular state. (By a 'state', we mean an observable situation, such as a particular number of particles in each of two boxes.)
As the numbers of particles increases, the number of possible arrangements in any particular state increases astronomically so we need a scaling factor to produce numbers that are easy to work with. Entropy, S, is defined by the equation
S = k lnW
where W is the number of ways of arranging the particles that gives rise to a particular observed state of the system. k is a constant called Boltzmann's constant which has the value 1.38 x 10-23 J K-1. In the expression S = k lnW it has the effect of scaling the vast number W to a smaller, more manageable, number.
ln is the natural logarithm, which also has the effect of scaling a vast number to a small one - the natural log of 10-23 is 52.95, for example.
Entropies are measured in joules per kelvin per mole (J K-1 mol-1). Notice the difference between the units of entropy and those of enthalpy (energy), kilojoules per mole (kJ mol-1).
The key point to remember is that entropy is a figure that measures randomness and, as you might expect, gases, where the particles could be anywhere, tend to have greater entropies than liquids which tend to have greater entropies than solids, where the particles are very regularly arranged, You can see this general trend from the animations of the three states.
Just as logarithms to the base 10 are derived from 10x, natural logarithms are derived from the exponent of the function ex, where e has the value 2.718. There are certain mathematical advantages to using this base number.
Don't let students worry about 'ln', just get them to use the correct button on their calculators. Some examples of calculating the ln of large numbers might help students to see the scaling effect.
The arrangement of particles in a solid, a liquid and a gas
|Substance||Physical state at standard conditions||Entropy, S
J K-1 mol-1
|Table 3: Some values of entropies|
observe that not all solids have smaller entropy values than all liquids nor do all liquids have smaller values than all gases. There is, however, a general trend that:
Ssolids < Sliquids < Sgasses