Energy and Human Ambitions on a Finite Planet, 2021a

B.4 Ideal Gas Law 381
B.4 Ideal Gas Law
Another topic covered in chemistry classes that strongly overlaps physics
is the ideal gas law. This relationship describes the interactions between
pressure, volume <strong>and</strong> temperature of a gas. In chemistry class, it is
learned as
PV nRT,
(B.2)
where P st<strong>and</strong>s for pressure (in Pascals 18 ), V is volume (cubic meters), n
is the number of moles, T is temperature (in Kelvin), <strong>and</strong> R is called the
gas constant, having the value
R 8.314
J
mol · K .
(B.3)
18: A Pascal (Pa) is also a Newton of force
per square meter, which reduces to more
fundamental units of J/m 3 (Joules of energy
per cubic meter).
To get degrees in Kelvin, add 273.15 (273 among friends) to the temperature
in Celsius. 19 St<strong>and</strong>ard atmospheric pressure is about 10 5 Pa. 20
20: 1 atmosphere is 101,325
19: And T( ◦ F) 1.8 · T( ◦ C) + 32.
Pa.
Example B.4.1 Let’s say we have a gas at “st<strong>and</strong>ard temperature <strong>and</strong>
pressure” (STP), meaning 0 ◦ C (273 K) <strong>and</strong> 1.013 × 10 5 Pa. How much
volume would one mole of gas 21 occupy?
We have everything we need to solve for volume, so
21: It may be surprising, but the ideal gas
law does not care what element or molecule
we are considering!
V nRT
P
(1 mol)(8.314 J/K/mol)(273 K)
1.013 × 10 5 Pa
≈ 0.0224 m 3 22.4L.
Okay; lots going on here. After the three values in the numerator are
multiplied, the only surviving unit is J (Joules of energy). The unit in
the denominator is Pascals, but this is equivalent to Joules per cubic
meter. So the answer emerges in cubic meters, as a volume should.
Since a cubic meter is 1,000 liters, we find that a mole of gas at STP
occupies 22.4 L—a number memorized by many a chemistry student!
Physicists prefer a variant of the ideal gas law that derives from the
study of “statistical mechanics,” which is practically synonymous with
thermodynamics <strong>and</strong> relates to the study of interactions between large
ensembles of particles. The form looks pretty familiar, still:
PV Nk B T.
(B.4)
Pressure, volume, <strong>and</strong> temperature are all unchanged, <strong>and</strong> expressed
in the same units as before. Now, N describes the number of particles
(quite large, usually), <strong>and</strong> k B is called the Boltzmann constant, having a
value
k B 1.3806 × 10 −23 J K . (B.5)
Notice that N, the number of particles, <strong>and</strong> n, the number of moles,
differs simply by a factor of Avogadro’s number, N A 6.022 × 10 23 .
Indeed, if we multiply N A by k B , we get 8.314, <strong>and</strong> are back to R. 22
22: The units work, too, since N A effectively
has units of a number (of particles) per mole.
© 2021 T. W. Murphy, Jr.; Creative Commons Attribution-NonCommercial 4.0 International Lic.;
Freely available at: https://escholarship.org/uc/energy_ambitions.