The road to the Boltzmann constant

If you’re going to think seriously about cellular biophysics, you’ve really got to clearly understand the Boltzmann constant and what it means.

The road to it starts well outside the cell, in the perfect gas law, part of Chem 101. This seems rather paradoxic. Cells (particularly neurons) do use gases (carbon monoxide, hydrogen sulfide, nitric oxide, and of course oxygen and CO2) as they function, but they are far from all gas.

Get out your colored pencils with separate colors for pressure, energy, work, force, area, acceleration, volume. All of them are combinations of just 3 things mass, distance and time for which you don’t need a colored pen,

The perfect gas law is Pressure * Volume = n R Temperature — the units don’t matter at this point. R is the gas constant, and n is the number of moles (chem 101 again).

Pressure = Force / Area
Force = Mass * Acceleration
Acceleration = distance / (time * time )
Area = Distance * distance

Volume = Distance * distance * distance

So Pressure * Volume = { Mass * distance / (time * time * distance * distance ) } * { distance * distance * distance }

= mass * distance * distance / ( time * time )

This looks suspiciously like kinetic energy (because it is )

Since work is defined as force * distance == mass * acceleration * distance

This also comes out to mass * distance * distance / ( time * time )

So Pressure * Volume has the units of work or kinetic energy

Back to the perfect gas law P * V = n * R * T

It’s time to bring in the units we actually use to  measure energy and work.

Energy and work are measured in Joules. Temperature in degrees above absolute zero (e.g. degrees Kelvin) — 300 is close to body temperature at 81.

Assume we have one mole of gas. Then the gas constant (R) is just PV/T or Joules/degree kelvin == energy/degree kelvin.

Statistical mechanics thinks about molecules not moles (6.022 * 10^23 molecules).

So the Boltzmann constant is just the Gas constant (R) divided by (the number of molecules in a mole * one degree Kelvin ) — it’s basically the energy each molecule posses divided by the current temperature — it is called k and equals 8.31441 Joules/ (mole * degree kelvin)

Biophysicists are far more interested in how much energy a given molecule has at body temperature — to find this multiply k by T (which is why you see kT all over the place.

At 300 Kelvin

kT is
4 picoNewton * nanoMNeters — work
23 milliElectron volts
.6 kiloCalories/mole
4.1 * 10^-21 joules/molecule — energy

Now we’re ready to start thinking about the molecular world.

I should do it, but hopefully someone out there can use this information to find how fast a sodium ion is moving around in our cells. Perhaps I’ll do this in a future post if no one does — it’s probably out there on the net.

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  • chemdiary  On January 11, 2016 at 9:31 am

    Looks like 200 sodium ions (max.) pass through a single sodium pump at any second.

    Membrane is ~4nm thick and if we assume the pump is also ~4nm, we can easily find it.

  • luysii  On January 11, 2016 at 10:24 am

    True, but they’re moving through in response to an electrical potential difference across the membrane, so their motion is more than thermal

    • chemdiary  On January 11, 2016 at 10:27 am

      You are right but I am not sure how much difference it makes. I am not a biophysicist or a physical chemist so I may be completely ignorant and wrong about it though.

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