pV^gamma=const

Google: adiabatic expansion gamma

For example:

I'll take a shot at answering, though the answer may or may not be definitive.

The way that equation is derived is from an equation with rates, that are then integrated. A constant in these circumstances means that what you end up with depends on what you started out with, in the same way as integrating an equation in accelerations provides an equation in velocity, with some constant, representing a beginning velocity to which some acceleration was applied.

The constant in this P.V^gamma = const case can be taken to represent an expression in entropy and strictly applies to isentropic situations.

Brian W

Well, it's not a universal constant; it depends on the conditions.

For example, we can model supersonic flow in a converging- diverging nozzle fairly well by assuming an ideal gas with constant specific heats. There is a range of exit pressures for which we have a normal shock wave in the nozzle. The shock itself is not isentropic, but the flows upstream and downstream are. This means, in particular, that

p v^gamma is constant on both sides of the shock wave, with two different constants.

vale, rip

gamma = c_p / c_v

c_p = specific heat at constant pressure c_v = specific heat at constant volume.

It can also be expressed as

m dp gamma = ---- --- R.T dr

m = molecular weight R = gas constant T = temperature p = pressure r = density.

For classical thermodynamics I always refer to Max Planck. One of the reasons is that he draws no line between physics and chemistry. Another is that he revised the book over twenty-five years. Finally, read the masters not their students.

Moving along and taking account of the kinetic theory of gases and statistical physics we find that

number of degrees of freedom gamma = ------------------------------ 2

Number of degrees of freedom is the number of bins for each molecule into we can distribute energy.

Enrico Fermi's monograph on thermodynamics covers these matters quite well.

That formula is incorrect. z = number of degrees of freedom

z + 2 gamma = -----. z

Yes, very much so. The problem is that gamma depends on the substance. The particulars can be found later in your thermo class (look at specific heats under constant pressure and under constant volume) and in stat mech classes. For a detailed description, it is necessary to look at the quantum mechanics of the molecules involved and gamma can then depend on temperature also.

Historical note: The value of gamma was predicted by Maxwell for monatomic and diatomic gasses accurately. However, when his theory was applied to more complicated molecules, it failed. This was one of the earliest signals that classical mechanics was wrong when applied to molecules.

--Dan Grubb

pV^gamma is only constant for adiabatic processes. gamma is the ratio of constant-pressure and constant-volume specific heats, which varies from gas to gas but is reliably predictable for monoatomic and diatomic gases. Most physics intro books for scientists and engineers will give several numeric examples.

PD

Others have made specific comments on the evaluation of the constant itself. For a bit more insight, you should consider that this is a special case of a reversible polytropic process for an ideal gas.

For many processes, if you were to plot log P vs. log V you would get a straight line, hence PV^n = constant

n=0 == isobaric n=1 == isothermal n= gamma == isentropic n= infinity ==isochoric and the work done is the integral of PdV or a constant times the integral of dV/(V^n).

or W = (P2V2-P1V1) / (1-n) = mR(T2-T1)/(1-n) for all but n=1.

Michael Press wrote in news: snipped-for-privacy@news.sf.sbcglobal.net:

That's right. Gamma = 1+1/n, where n = # degrees of freedom / 2, known as the "polytropic index."

rip pelletier wrote in news: snipped-for-privacy@news.supernews.com:

Thanks. I did not know that. ?:] Maybe we can get a hint about the constant from dimensional analysis. Do you know what are the units of V^gamma?

Well, since the units of volume are distance^3, I would expect the units of V^gamma to be distance^(3*gamma).

Seriously, the equation p*V^gamma =3D constant is valid ONLY for adiabatic processes of an ideal gas.

The inclusion of the ratio of heat capacitues allows it to hold for poly-atomic gases.

Also, the applicability is limited because gamma is NOT a constant. The heat capacities vary with temperature.

The heat capacities have contributions from translation (the 'ideal gas' motion), from rotation (due to the spinning of non-spherical molecules_ and from vibration (internal motions within the molecule), and at higher temperatures from electronic excitations.

Each of these motions makes an additive contribution to the heat capacity. The vibrational motion is a particularly complex function of temperature.

In adiabatic processes in which temperature does not change very much (so the heat capacities are relatively constant over the range of temperatures of interest) gamma can be treated as a constant, so

p*V^gamma =3D constant

is approximately true. Note that the expression also depends on the AMOUNT of material present.

Tom Davidson Richmond, VA

" snipped-for-privacy@comcast.net" wrote in news: snipped-for-privacy@w35g2000yqm.googlegroups.com:

The adiabatic approximation is a very good one, valid even in a highly turbulent regimes, and only breaks down in sonic flows.

Gamma's variability with temperature is a STEP FUNCTION, unrelated to non-adiabaticity. Rotational and vibrational modes depend on molecular geometry, and only appear above the lowest mode's energy threshold, well above room temperature. In this case gamma is merely a different constant, than the value of 5/3 that you're used to. Gamma = 7/5 is the next simplest value, in the diatomic case, and this is where the units of pV^gamma cease to make sense. It it works out nicely if gamma=5/3, but not otherwise. Gamma=1+2/n always, where n=#degs. of f'dom.

On Sun, 01 Mar 2009 11:50:04 GMT, in sci.engr.mech Bluuuue Rajah wrote:

Nonsense the functional temperature dependence of the ratio of specific heat of gases is not a step function and has an important application in high speed gas dynamic calculations. See for instance,

and

Computation of Thermally Perfect Compressible Flow Properties Author(s): Witte, David W.; Tatum, Kenneth E.; Williams, S. Blake Abstract: A set of compressible flow relations for a thermally perfect, calorically imperfect gas are derived for a value of c(sub p) (specific heat at constant pressure) expressed as a polynomial function of temperature and developed into a computer program, referred to as the Thermally Perfect Gas (TPG) code. The code is available free from the NASA Langley Software Server at URL The code produces tables of compressible flow properties similar to those found in NACA Report 1135. Unlike the NACA Report 1135 tables which are valid only in the calorically perfect temperature regime the TPG code results are also valid in the thermally perfect, calorically imperfect temperature regime, giving the TPG code a considerably larger range of temperature application. Accuracy of the TPG code in the calorically perfect and in the thermally perfect, calorically imperfect temperature regimes are verified by comparisons with the methods of NACA Report 1135. The advantages of the TPG code compared to the thermally perfect, calorically imperfect method of NACA Report 1135 are its applicability to any type of gas (monatomic, diatomic, triatomic, or polyatomic) or any specified mixture of gases, ease-of-use, and tabulated results. ber: AIAA Paper 96-0681

snipped-for-privacy@w35g2000yqm.googlegroups.com:

The heat capacities of monatomic and diatomic gases are approximately at the limiting values predicted by the equipartition principle at 300 kelvins and higher. That makes Gamma =3D 1 + 2/n, approximately.

Not so for more complex molecules:

Tom Davidson Richmond, VA