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Re: [Phys-L] House Heating

On 11/14/20 1:46 AM, Bob Sciamanda via Phys-l wrote:

Consider a house filled with air of a monatomic ideal gas, shared with the
outside atmosphere, so that PV = NkT can model the house air. The gas
energy is E = (3/2)NkT.

1) Remark: This is a classic illustration of the difference between
energy and temperature.

2) Taking a further step down that road, we need not assume the gas
is monotonic. Instead let's assume it is /polytropic/ with some
adiabatic exponent γ (gamma).

P V = N kT = (γ - 1) E

Note that
γ = 5/3 for a monatomic gas
γ = 7/5 for a diatomic "rigid rotor" gas

Perhaps easier to remember, and closer to the fundamental physics, is:

E = P V / (γ - 1)
= D▯ P V / 2 (sometimes!)

where D▯ (pronounced D quad) is the effective number of *quadratic*
degrees of freedom per particle:
D▯ = 3 for a monatomic gas
D▯ = 5 for a diatomic "rigid rotor" gas

Of course not everything is quadratic, in which case trying to count
the degrees of freedom doesn't tell you what you need to know ... but
even then, the equation of state can be *locally* approximated as
polytropic, with "some" value of γ.

For ordinary air near room temperature, γ = 1.4 is a better approximation
than γ = 1.666.

All-too-often people talk about degrees of freedom without being clear
about what's quadratic and what's not. For example, a particle in a 1D
parabolic potential well (harmonic oscillator) has D▯ = 2 (one potential
plus one kinetic) whereas a particle in a square-well potential (particle
in a box) has only D▯ = 1 (kinetic only). The potential degree of freedom
is still there for the particle in a box, but it doesn't affect the γ in
the same way, because it's not quadratic.


Tangentially related: P V has dimensions of energy, and all-too-often
students assume "the" energy must be equal to P V, when in fact it is
quite a bit less than that. Lots of things have dimensions of energy
but are not "the" energy. There is more to physics than dimensional