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*From*: John Denker <jsd@av8n.com>*Date*: Sat, 14 Nov 2020 06:48:04 -0700

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

analysis.

**Follow-Ups**:**Re: [Phys-L] House Heating***From:*Brian Whatcott <betwys1@sbcglobal.net>

**References**:**[Phys-L] House Heating***From:*Bob Sciamanda <treborsciamanda@gmail.com>

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