The quantum casino: tutorials |

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 ln*W*

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 ln*W *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 10^{x}, natural logarithms are derived from the exponent of the function e^{x}, 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 10^{23} 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), *kilo*joules 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, SJ 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.

Entropy |