Chronology |
Current Month |
Current Thread |
Current Date |

[Year List] [Month List (current year)] | [Date Index] [Thread Index] | [Thread Prev] [Thread Next] | [Date Prev] [Date Next] |

*From*: John Denker <jsd@av8n.com>*Date*: Sun, 17 Jun 2007 14:39:43 -0400

On 06/17/2007 12:19 PM, Jeffrey Schnick wrote:

I have a question about Equation 45.17 in the Feynman Lectures on

Physics.

Excellent question. You know folks are on the ball when

they find bugs in the Feynman lectures.

We shall see that the difficulty here is traceable to being

sloppy about what is held constant and what is not. This

seems to be a raging disease wherever thermodynamics is

discussed; not even Feynman is immune.

Feynman is applying thermodynamics to photons and has chosen

to treat T and V as independent variables and to treat U(T,V) and P(T,V)

as dependent variables.

Yes, that's what he /says/ he will do ... but no, that's

not what he really /does/ do. See below.

He takes the partial derivative of U=3PV (comes

from equation 39.17) holding T constant and obtains:

dU/dV (const T) = 3P

I was expecting:

dU/dV (const T) = 3P + 3V dP/dV (const T)

That expectation is correct.

Why is the second term missing? Its absence would suggest that P(T,V)

is actually only a function of T, P(T).

For photons under black-body conditions, P(T,V) is independent

of V. You might have surmised as much from a scaling argument:

We know P is intensive and T is intensive, so if there are no

other variables involved, you can't have an extensive variable

on one side of the equation and not on the other.

Remember: The idea that XXX is an intensive property is just

a particularly simple scaling property: It means that XXX

scales like the size of the system to the zeroth power.

Interestingly enough, for ordinary gas molecules under "conventional"

conditions, it is not true that P is independent of V in this

way.

Feynman is pulling a fast one here, because there is actually

another variable running around, namely N, the number of photons.

When dealing with ordinary gas molecules, it is /conventional/

to assume that changes in T and/or V take place at constant N

unless otherwise stated. This is called a canonical ensemble.

When dealing with photons, it is hard to keep N constant over

long or even medium-long periods of time. It turns out that

Feynman is not saying

dU/dV (const T,N) = 3P

but rather

dU/dV (const T,mu) = 3P

where mu is the chemical potential. This is called a

grand canonical ensemble.

This is a trap for the unwary, due to conflicting conventions.

You can reconcile these two ideas by imagining that

a) you expand the photon gas at constant N for a very short time,

then

b) pause while little antennas in the walls radiate some

more photons into the box.

Step (a) is the same for molecules as for photons. Step (b)

is unconventional but not impossible for molecules; imagine

a reservoir of high-density liquid that evaporates so as to

make the gas density a function of (T) and not (T,V).

Actually there is another somewhat-related trap here. The

pressure is -- as always -- defined in terms of an isentropic

expansion, but here we are considering an isothermal expansion.

You can reconcile these ideas by imagining that the pressure

measurement is done sufficiently quickly that it is isentropic,

while the expansion is done slowly enough that it is isothermal.

Equation 39.18, PV^(4/3)=C

(where C is a constant), would lead one to believe that P is indeed a

function of V.

This can be considered another outbreak of the same disease:

being sloppy about what is held constant and what is not.

The law that P V^gamma = constant is not universally true;

it applies only to isentropic processes. To remind myself

of this, I prefer to write

P V^gamma = f(S)

where S is the entropy, and f(S) is "some function" of the

entropy.

You know I like to visualize these relationships; in this case

the picture is that the contours of constant (P V^gamma) are

everywhere parallel to the contours of constant entropy.

**Follow-Ups**:**Re: [Phys-l] Photon Thermodynamics***From:*"Jeffrey Schnick" <JSchnick@Anselm.Edu>

**Re: [Phys-l] Photon Thermodynamics***From:*carmelo@pacific.net.sg

**References**:**[Phys-l] thermo differential and extensive/intensive variables***From:*Stefan Jeglinski <jeglin@4pi.com>

**Re: [Phys-l] thermo differential and extensive/intensive variables***From:*John Denker <jsd@av8n.com>

**Re: [Phys-l] thermo differential and extensive/intensive variables***From:*Stefan Jeglinski <jeglin@4pi.com>

**Re: [Phys-l] thermo differential and extensive/intensive variables***From:*John Denker <jsd@av8n.com>

**Re: [Phys-l] thermo differential and extensive/intensive variables***From:*Stefan Jeglinski <jeglin@4pi.com>

**[Phys-l] Photon Thermodynamics***From:*"Jeffrey Schnick" <JSchnick@Anselm.Edu>

- Prev by Date:
**[Phys-l] Photon Thermodynamics** - Next by Date:
**[Phys-l] Are Concepts Instantiated in Brain Synapses? (was Bunkum Awards #2. . . .)** - Previous by thread:
**[Phys-l] Photon Thermodynamics** - Next by thread:
**Re: [Phys-l] Photon Thermodynamics** - Index(es):