exemplified When You Heat Your House Does the Thermal Energy Content Increase? Illustrating the first l a w of thermodynamics
Contribution by Zayn Bilkadi, University of Rochester I n a short letter' t o the British journal Nature entitled "Why do we have winter heating?" the noted astrophysicist Robert Emden suggested that winter heating does not lead t o an increased thermal energy. We quote the first paragraph of that letter Why do we need winter heating? The layman will answer "To make the room warmer;" the student of thermodynamim will perhaps so express it: "To impart the lacking (inner, thermal) energy." If so, then the layman's answer is right, the scientist's is wrong.
Even though Emden's reasoning was essentially right, a criticism t,o his paper was given by Arnold Sommerfeld later. We propose in the following to show first how Emden arrived at his conclusion, then we will add Sommerfeld's correction. Emden starts from the reasonable assumption that the pressure inside the room being heated is equal to the pressure outside it. The (inner, thermal) energy per unit mass of air is ZL
=
euT
where c, is the beat capacity per unit mass of air at constant volume, T being the absolute temperature of the room. Then the energy density in the room (i.e., thermal energy per unit volume) is ul
=
c.pT
of claret from the cold cellar and put it in the warm room. It becomes warmer, but the increased energy content is not horroved from the air of the room but is brought from the outside(!)." Arnold Sommerfeld2accepted Emden's idea hut corrected him in his statement that the energy density in the room is constant. Sommerfeld felt that the energy density in the room should decrease as the room is being heated. He preferred to m i t e the energy as u
=
U, =
ut = c,PM/R
For a pressure of 1atm u, = 0.0604 cal c ~ n - ~ . Emden hence concluded that the thermal energy content of the room is constant, independent of temperature, and will remain so no matter how much heat we pour into the room. All the thermal energy that is added to the room escapes through the pores of the walls. The author then continues: "I fetch a bottle
pu
Therefore Since pT
= M P I R , one may write UI =
+ duo - cuTo)
cUMP/R
The second term on the right hand side of this last relation could be shown to bc positive and greater in magnitude than the first term. The first term on the right hand side of the last relation is temperature independent but the second term is not, because p is inversely proportional t o the temperature; hence as the temperature increases, the second term on the right decreases, and consequently the energy density decreases. Thus Sommerfeld concluded that the energy density decreases as the temperature increases.
Some further insights by Professor Wilbur B. Bridgman Worcester Polytechnic Institute
RpT/M
where M is the molecular weight. Hence
- To)
Where Todenotes some temperature above the point of liquefaction of the gas (air) at ~ ~ h i cthe h ideal gas law is still valid. The energy density u~is related to the energy per unit of mass, u, by therelation
where p is the density of air. Assuming ideal behavior of the air in the room P
- uo = c,(T
Some interesting thermodynamic relations are involved in considering the questions raised above. Rooms are ordinarily so constructed that the heating process is best considered as being at constant pressure. Thus the quantity of heat required to raise the air temperature is differentially defined by where n is the initial number of moles of gas in the room and is the molar heat capacity a t constant pressure. In terms of the first lan- we can also say that
cp
' EMDEN, R., Nature, 141, 908 (1938). SOMM~RFELD, A., "Thermodynamics and Statistical Mechanics," Academic Press, New York, 1964, p. 357. a
For an ideal gas this becomes Volume 49, Number 7, July 1972
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493
69 = nCvdT
+ nRdT
(3)
Because the increase in temperature causes expansion, some of thc air is forced to leave the room. By application of the ideal gas law, n', the number of moles of gas remaining in a room of volume, V ,is given by
v
T nr = v + d v n = M n
(4)
and nu,the number of moles that leave the room is therefore
By combining cqns. (3), (4), and (5) Sy =
T
dT d v d T
+
+dT- nCvdT + nRdT T + dT
(6)
Since CV = 3R/2 for monatomic gases and all other gases have even larger heat capacities, wc conclude that the first tcrm in cqn. (6) is thc largest and the second tcrm the smallest. The first term represents thc increase in intcrnal cncrgy of the gas molecules that remain in the room. Thc second term represents that fraction of the addcd encrgy that is carried out of the room by the cxpansion of the gas. The last term represents the portion of the addcd encrgy used to perform the ncccssary cxpansion work. Whila the escaping molecules carry off only a small fraction of t,hcheat increment used to raise the temperature by dT, they arc carrying with them the intcrnal energy t,hat they possess at tcmperature T. This may bc a much marc significant quantity, as shown by the following argument. Initially thc total internal energy in the room can be represented as where I?, is the internal energy per mole at temperature T. Similarly Combining eqns. (4), (7), and (8) dE,..,
=
n'(&
CVT)~T + G ~ T- )nEr = n(-ET1' ++ dT
This equation indicates that the sign and magnitude of the change in internal energy depends on the difference betwccn ET and C V ~ Any . conclusions that are drawn from eqn. (9) must be based upon assumptions regarding the content and zero reference point of ET. In ordcr to compare the results of previous authors and discuss the various viewpoints, let us write ET = E*
+ C"T
(10)
The C,T term may be interpreted as representing that portion of the internal energy that depends directly on the thermal motion of the molecules. For an ideal monatomic gas where CV is 3R/2, this is consistent with kinetic theory that relates 3RT/2 to the translational kinetic energy of the molecules. For more complex molccules where Cv is greater than 3R/2, contributions from rotational and vibrational motions of the molecules ire included also. E* can then he interpreted as a constant that includes all contributions to the internal energy of the gas that are independent of the gas tcmperature. Chemical bonding energy, nuclear energy, energy increments associated with phase 494
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Journal of Chemical Education
changes are among the effects that could he included in this term. The assignment of a zero reference point also affects its magnitude. When eqn. (10) is substituted into eqn. (9), the term in parentheses becomes simply -E*. If E* is zero, then dE,, is zero. This corresponds to the conclusion of Emden1that the energy content of the room remains constant. A zero value for E* may he interpreted as a reference state consisting of molecules that have no thermal energy (O°K) and no intermolecular interactions (infinite separation). This reference state is often implied in applications of the kinetic theory to gases. By using this reference point, consideration is effectively limited to only the thermal energy represented by C"T. When one considers only this thermal component of the internal energy inherently present in the escaping molecules, one concludes that they carry out of the room an amount of thermal energy exactly equal to the increase in energy experienced by the remaining molecules. On the other hand, SommerfeldZreached the conclusion that the internal energy of the room decreases as temperature increases. This corresponds to assigning E* a positive value. This can result from assigning the reference state to the normal solid substance at O0IC Sublimation energy must he supplied to the molecules in this reference state to put them in a gaseous condition corresponding to the previous reference condition. From this viewpoint a positive value of E* seems reasonable. With this contribution to the internal energy included, it appears that the energy carried off by the escaping molecules is greater than the increase in energy within the room. Another widely used reference point for energy calculations is infinite separation of elementary particles. If the elementary units arc taken to be atoms, then for monatomic gases this would he equivalent to the first case. For diatomic and more complex molecules, the exothermic process of forming molecules from atoms leads to a negative value for E*. If subatomic particles (electrons, nuclei, protons, neutrons, etc.) were used as the reference state, E* would become increasingly negative. Any definition that makes E* negative leads eqn. (9) to predict a net increase in the internal energy of the room. All of this discussion emphasizes that thermodynamics can be used to accurately evaluate changes in thermodynamic quantities hut not their absolute value. We can be confident that most of the energy used to heat the air in the room is used to increase the thermal energy of the gas molecules in the room. Only a small fraction of this added energy is carried away by the escaping gas. However, what the net change in the total energy content of the gas in the room is, is not uniquely defined. Different conclusions are reached depending on the choice of reference state. An individual's feeling of comfort or discomfort in the room depends primarily on the relatiocship betu7een room temperature and his body temperature. He is directly affectedby the temperature at which the energy in the surrounding air is available, not by the quantity of energy. Thus we heat our buildings in winter to make energy available at a higher temperature. Whether or not the total energy content of the air increases or decreases cannot be answered unambiguously.
