Talk:Sackur–Tetrode equation

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[edit] Various points

Hi PAR. (1) Why use E for internal energy when the article on ideal gas uses U? (2) Why not define u=U/kN and v=V/kN for specific energy and volume ? (3) The argument to the logarithm seems not to be dimensionless! Bo Jacoby 13:06, 9 November 2005 (UTC)

Good questions -

• U is the correct letter to use according to a IUPAC article and I will change it - Alberty, R. A. (2001). "Use of Legendre transforms in chemical thermodynamics". Pure Appl. Chem. Vol. 73 (8): 1349–1380. 
• The reason I use U and V is probably the same reason you use u and v. Its the way I learned the subject, and I don't see an inherent advantage to either notation.
• The requirement that the argument of a logarithm be dimensionless can be relaxed somewhat since ln(x)+ln(1/x) is dimensionless, even when x is not. If you combine all the logarithms, it should be dimensionless.

Which brings up a point that worries me, and it is not a point for someone who is book-bound, so I know I'm talking to the right person. Even though its not generally done, you should be able to assign the additional dimension of "particle" to certain quantites. N has dimension of "particles" so that N/V has dimensions of "particles per unit volume". Boltzmanns constant k has units of "entropy per particle", so that kT is "energy per particle". Planck's constant has units of "action per particle". I cannot get the argument of the logarithm in the Sackur-Tetrode equation to be dimensionless doing this. The idea that particle=dimensionless is just a mindless hand-me-down rule that I cannot figure out how to disprove nor justify. I have a strong hunch that its ok to assign the dimension "particle" and that my inability to render the argument dimensionless points out that I am missing some subtle point in the physics. PAR 16:55, 9 November 2005 (UTC)

I am not an expert on the Sackur-Tetrode formula, and it confuses me. I would expect it to asymptotically approach the ideal gas formula, but that is not evident to me. Boltzmanns constant k is joule per kelvin per molecule because PV/T = kN. I am not sure whether pressure should be called P or p. Personally I prefer the big letter, because V and T are big letters, but the article on pressure uses the small letter. Plancks constant h is joule per hertz per particle because E = Nhν. Not only is the unit 'particle' usually omitted, but the unit of angle, turn, is also omitted, leaving confusion on whether it is joule second per turn per particle or joule second per radian per particle, (h-bar). I have had big difficulties in understanding thermodynamics, mostly because the units were messy. (The natural gas industry expresses the gas content of crude oil in normal cubic feet per barrel). There is no name for the SI-unit of entropy, the joule per kelvin. I suggest to call it a clausius. It is also the unit of matter, and of heat capacity. Nor is there a single letter signifying amount of matter: nR=kN=?. I would prefer to get rid of the mol. I prefer to express matter density in clausius per cubic meter, which is the same as pascal per kelvin. In short: I agree that good use of units clarifies science. Bo Jacoby 14:12, 11 November 2005 (UTC)

The ideal monatomic gas entropy is S / Nk = ln(VT^{3 / 2} / NΦ) where Φ is some undetermined constant. The Sackur-Tetrode equation specifies that constant, so the two are completely compatible.

That's nice. But changing Φ merely changes the zero point of entropy. S / Nk = ln(V / Nk) + (3 / 2)ln(T) − ln(Φ / k). So what is the physical significance of the formula ? Bo Jacoby 13:32, 14 November 2005 (UTC)

I've never need to use it practically, so I'm sort of winging it here - If you are dealing with entropy differences, you dont need to know the constant. If you are dealing with enormous entropies (S/Nk huge) then again, no need. If you are dealing with absolute entropy at or near the critical point (S/Nk of order unity) then still no need, it breaks down. But for S/Nk in an intermediate range, the question of what is the constant is important. Check this out.

I think there should be a strong distinction between the ideas of dimensions and units. The speed of light has the dimensions of velocity or distance/time, and has units of m/sec in the SI system, cm/sec in cgi system, feet/sec in Imperial units. It can also be measured in furlongs/fortnight. From a theoretical point of view, who cares about the units? The dimensions are of fundamental theoretical importance, the units are not (except that they tell you the dimensions.) Worrying about units is like worrying about the language a scientific paper is written in. Who cares? as long as its translated into a language you understand. Worrying about dimensions is like worrying about what the paper is saying, and theoretically, this is worth worrying about. Worrying about units is of little theoretical significance (but of huge practical significance, of course.)

The bottom line is that units are vitally important to communication, just as language is. I don't have a favorite language, but I do have one that I am confined to speak in because of my upbringing and mental limitations. I don't wish to be similarly confined by having a "favorite" set of units. Units are just some language I have to learn in order to communicate with others. Dimensions are much more interesting. PAR 16:30, 11 November 2005 (UTC)

If an author sticks to one system of units, then he needs not distinguish between dimension and unit, because for each dimension there is but one unit in a well designed system of units. (If an author sticks to one language, then he needs not distinguish between concept and word either.) Bo Jacoby 13:32, 14 November 2005 (UTC)

Yes, the distinction will never be a problem if you live on a desert island. In reality, you have to negotiate the difference between concept and word with other people, and to do so effectively you need to understand the difference between the two. In my mind, I try to deal with concepts. The process of translating these concepts to words is extremely negotiable in my mind, whatever it takes to communicate. Which means I have little respect for "proper english" while at the same time I strive to be adept at it. I have little respect for units either, yet I always try to get them right.

Thats why I like the topic of dimensional analysis especially the Buckingham Pi theorem - one of its basic tenets is that all physical theories must be expressible in dimensionless terms in order to have any validity. That says it all! PAR 17:04, 14 November 2005 (UTC)

I too like dimensional analysis. The logarithm of the speed of light is log(300000ms^-1)=5+log(3)+log(m)-log(s). The logs of units define a basis of a vector space. Changing basis is changing units. The dimension length is the line {log(x)+log(m) | x in R}. The dimension area is the line {log(x) + 2log(m) | x in R}. The basis of a vector space is more basic than these lines. Thank you for pointing my attention to the Buckingham Pi theorem. Note that it can be expressed in units as well as in dimensions. Bo Jacoby 15:22, 15 November 2005 (UTC)

If you go back to the pre-SackurTetrode equation(leave out the N in V/N) then you get a dimensionless argument for the logarithm. Reason being:the argument is the ratio of two phase-space hypervolumes which yields the number of microstates consistent with the macrostate description. Dividing by N! spoils this. I don't believe the problem arises in quantum statistics because there you deal with the number of states from the outset.--GRaetz 17:25, 21 January 2006 (UTC)

There is a problem with the Gibbs paradox derivation as well. You have two conditions holding in the 6N dimensional phase space - A fixed energy:

and the gas is contained in a box:

where i is the i-th particle, and j=1,2,3 corresponding to x,y,z. U is energy, m is particle mass, p is momentum, x is position, L is the length of the edge of the box containing the gas. The first condition specifies the (3N-1)-dimensional surface of an 3N-dimensional sphere in the momentum part of the space, while the second specifies a 3N-dimensional volume in the space part. A 6N-dimensional volume in this phase space should have dimensions of (px)^3N or action^3N. The product of the above surface and volume does not give such a volume in phase space, and you can't divide by h^3N and get a dimensionless number. I've looked at Huang, and he (mistakenly, I believe) uses the volume of the hypersphere, rather than its surface area. Maybe I have something wrong, but I would like to know what it is. PAR 01:14, 22 January 2006 (UTC)