UPPER LIMIT of TEMPERATURE CARL ROSENBLUM Old Town Ribbon and Carbn Company, Inc., Brooklyn, New York
W
HEN Charles's Law is taken under consideration in the classroom and it is pointed out that because of the proportional decrease of volume with temperature, the lowest possihle temperature is about -273'C., the student naturally asks, "What is the highest possible temperature?" A search of textbooks reveals no answer to this problem. Therefore, the teacher is thrown on his own resources and may answer honestly, "It is probably very high, but I don't know how high," or be sarcastic and say, "What's the matter, isn't i t hot enough for you?" or resort to some of the other escapes which have been used. By the use of two equations and some simple mathematics, a value is obtained which may logically be considered the upper temperature limit. That there is such a limit seems possihle because, although the known temperature of stars goes up to millions of, degrees, these high temperatures seem to cluster and are not scattered haphazardly in astronomical figures on the temperature scale. A few of these high temperatures are given in the table below: T ~ K ~ ~ ~ A T U R B O X S T ~ R S ~
slor C~pella
sgccrrum
GO
."
r.nhni --Fa-
PO
V Puppis
B1 GO MO
sun
Krucger 80
surfocc kmpcrorurc 5,20O0K. s mnox -,-"" 19,000"K. B,OOO'K. 3,100"~.
c c n t d trmg~orurc 9,000,000°K. ti nnn nonor
42,000,000"K~ o,~oo,o~o'K. ~Z,OOO,OOO'K.
An upper limit may he obtained mathematically by considering the nature of heat and its effect on molecules. The molecules of a subst8nce are not a t rest. At ordinary temperature, the molecules of a gas in air move a t about '/a mile per second. As the temperature increases, so does the molecular velocity. The relationship between the velocity of the molecules and temperature may be obtained from the kinetic equation of a gas and the ideal gas law. The kinetic equation is:
where m is the mass of a single molecule, n is the number of molecules, u is the mean velocity in cm./sec., P is
-
' BAKE^, "Astronomy," 2nd ed., D. Van Nostrand Co., New
York City, 1933, p. 420.
the pressure (force exerted upon a unit area), and V is the volume of a cube containing the molecules. The ideal gas law is: PV
=
nRT
(2)
where R is a constant having the value of 8.31 5 X lo7 ergs per mol per degree Kelvin and T is the temperature in degrees Kelvin. For one mol equation (2) may be rewritten: PV
=
(3)
RT
From equations (1) and (3) we get: and since we are considering one mol, mn equals the molecular weight, M RT = '/~Mnu'
(5)
R is a constant having the value given above. Our formula in (5) becomes (solving for T) : This is the relationship of temperature and velocity of molecules. ~ s s u m i n ga molecular weight of 1, i i we find a limifing value for .u w6 have a limiting value for T. It has been considered that the s ~ e e dof livht. which is about 186,360 miles per second, is the maxi: mum speed of any obiect. Therefore, if we substitute this speed for u we obtain the upper limit of temperature, about 3.58 X 1012 degrees Kelvin, or, roughly, 3,580,000,000,000 degrees. Of course, to be accurate, this is the upper limit only for atomic hydrogen. If we use any other substance of higher molecular weight our temperature limit would be higher. However, it is known that in regions of high temperature, the lower molecular weights predominate. Therefore, we assume that should we approach the .upper limit not only would the higher molecular weights he broken down but we would have a predominance of protons. I n order for the upper temperature limit t o be reached, all the molecular energy would have to he transformed into heat. This is not true in nature, as a great deal of the energy is dissipated in the form of light, both visible and invisible, and in other forms of radiation. ~
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