## Entropy

**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 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

*it has the effect of scaling the vast number*

*S*= k ln*W**to a smaller, more manageable, number.*

*W**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),

*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 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.

### Note

Just as logarithms to the base 10 are derived from *10 ^{x}*, natural logarithms are derived from the exponent of the function

*e*, where

^{x}*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, 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 |

### Note

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:

* S_{solids}* <

*<*

*S*_{liquids}

*S*_{gasses}**Further values may be obtained from the RSC Electronic Data Book**