Entropy

Scientists have a mathematical way of measuring randomness – it is called entropy and it 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.)

 Boltzmann’s constant is a measure of the amount the energy of a single particle (such as an atom or molecule) of a gas increases for a 1 K increase in temperature.

As the numbers of particles increase, 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.

 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 advantaged to using this base number.

ln is the natural logarithm, which also has the effect of scaling a vast number to a small one – the natural log of 1023 is 52.95, for example. (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.)

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 have greater entropies than liquids which tend 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.

 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 Carbon (diamond) solid 2.4 Copper solid 33 Calcium oxide solid 40 Calcium carbonate solid 93 Ammonium chloride solid 95 Water (ice) solid 48 Water liquid 70 Water (steam) gas 189 Hydrogen chloride gas 187 Ammonia gas 192 Carbon dioxide gas 214

Table 3 Some values of entropies
Further values can be obtained from The RSC Electronic Data Book

Note that not all solids have smaller entropy values than all liquids nor all liquids smaller values than all gases.

 Home Videos Tutorials The direction of chemical reactions Entropy The Second Law of Thermodynamics Is the Second Law wrong? The role of energy The system and the surroundings Total entropy change The Gibbs free energy ∆G and temperature Reversible reactions Why is free energy 'free'? Simulations Entropy and disorder Arranging objects The distribution of energy The approach to equilibrium Equilibrium mixtures ∆G and temperature 