# The Universe of Discourse

Tue, 16 Feb 2021

Katara is toiling through A.P. Chemistry this year. I never took A.P. Chemistry but I did take regular high school chemistry and two semesters of university chemistry so it falls to me to help her out when things get too confusing. Lately she has been studying gas equilibria and thermodynamics, in which the so-called ideal gas law plays a central role: $$PV=nRT$$

This is when you have a gas confined in a container of volume !!V!!. !!P!! is the pressure exerted by the gas on the walls of the container, the !!n!! is the number of gas particles, and the !!T!! is the absolute temperature. !!R!! is a constant, called the “ideal gas constant”. Most real gases do obey this law pretty closely, at least at reasonably low pressures.

The law implies all sorts of interesting things. For example, if you have gas in a container and heat it up so as to double the (absolute) temperature, the gas would like to expand into twice the original volume. If the container is rigid the pressure will double, but if the gas is in a balloon, the balloon will double in size instead. Then if you take the balloon up in an airplane so that the ambient pressure is half as much, the balloon will double in size again.

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I had seen this many times and while it all seems reasonable and makes sense, I had never really thought about what it means. Sometimes stuff in physics doesn't mean anything, but sometimes you can relate it to a more fundamental law. For example, in The Character of Physical Law, Feynman points out that the Archimedean lever law is just an expression of the law of conservation of energy, as applied to the potential energy of the weights on the arms of the lever. Thinking about the ideal gas law carefully, for the first time in my life, I realized that it is also a special case of the law of conservation of energy!

The gas molecules are zipping around with various energies, and this kinetic energy manifests on the macro scale as as pressure (when they bump into the walls of the container) and as volume (when they bump into other molecules, forcing the other particles away.)

The pressure is measured in units of dimension !!\frac{\rm force}{\rm area}!!, say newtons per square meter. The product !!PV!! of pressure and volume is $$\frac{\rm force}{\rm area}\cdot{\rm volume} = \frac{\rm force}{{\rm distance}^2}\cdot{\rm distance}^3 = {\rm force}\cdot{\rm distance} = {\rm energy}.$$ So the equation is equating two ways to measure the same total energy of the gas.

Over on the right-hand side, we also have energy. The absolute temperature !!T!! is the average energy per molecule and the !!n!! counts the number of molecules; multiply them and you get the total energy in a different way.

The !!R!! is nothing mysterious; it's just a proportionality constant required to get the units to match up when we measure temperature in kelvins and count molecules in moles. It's analogous to the mysterious Cookie Constant that relates energy you have to expend on the treadmill with energy you gain from eating cookies. The Cookie Constant is !!1043 \frac{\rm sec}{\rm cookie}!!. !!R!! happens to be around 8.3 joules per mole per kelvin.

(Actually I think there might be a bit more to !!R!! than I said, something about the Boltzmann distribution in there.)

Somehow this got me and Katara thinking about what a mole of chocolate chips would look like. “Better use those mini chips,” said Katara.