Do intramolecular vibrations contribute to temperature?
Thanks,
Rob
Um, some have described collisions between molecules to answer this. That may be true of a gas…but temp of a solid? Do molecules hit against each other in a solid? These collisions being described relate to enthalpy. Yet, can’t two solids have the same enthalpies and different temps? For solids, it depends on the specific heats of different materials. Remember H = mcdT? Where H = enthalpy, in BTUs. (Some may have called it Q.) Or, some may know it as h =cdT where h = enthalpy/lb (or q in the same units on a per pound basis.)
Just a minor point here; you seem to be suggesting that collision frequency in a gas is small enough that collisions can be considered ‘occasional’; but in fact at atmospheric pressure and temperature, collision frequency is of the order of 10^9 / s. The mean free path (between collisions) is about 1000 times the molecular diameter.
I’ve always found it useful to think in terms of molecular separation as well - solid: molecules ‘touching’ and held in place; liquid: molecules ‘touching’ but able to slide past each other; gas: molecules widely separated but still interacting through collisions.
Like I said, it’s a minor point, but it did help me visualise why gases are highly compressible, but solids and liquids resist compression.
Thanks for the correction… that does help clarify the picture. And it’s good to hear that my explanation was pretty much on target except for that.
Molecules in a solid, in spite of our macroscopic experience, are wildly in motion at their level. I think you also have electrons moving to higher energy levels, but I’m not the guy to ask about that. When you heat a piece of steel, for example, those collisions are pretty frequent. When you heat it to red hot, the electrons are going to higher energy levels and then emitting photons when they drop back down (I think). The secret life of solids.
Two materials (liquids, solids, whatever) can have the same enthalpy but different temperatures, depending on their specific heat. In, fact, you asked the question so thoroughly it seems that the answer is right in your question.
KelvinBaron Kelvin of Largs (William Thomson, 1824-1907)
On an Absolute Thermometric Scale founded on Carnot’s Theory of the Motive Power
of Heat[1], and calculated from Regnault’s Observations[2]
Phil. Mag. October 1848 [from Sir William Thomson, Mathematical and Physical
Papers, vol. 1 (Cambridge University Press, 1882), pp. 100-106.]
The determination of temperature has long been recognized as a problem of the
greatest importance in physical science. It has accordingly been made a subject
of most careful attention, and, especially in late years, of very elaborate and
refined experimental researches[3]; and we are thus at present in possession of
as complete a practical solution of the problem as can be desired, even for the
most accurate investigations. The theory of thermometry is however as yet far
from being in so satisfactory a state. The principle to be followed in
constructing a thermometric scale might at first sight seem to be obvious, as it
might appear that a perfect thermometer would indicate equal additions of heat,
as corresponding to equal elevations of temperature, estimated by the numbered
divisions of its scale. It is however now recognized (from the variations in the
specific heats of bodies) as an experimentally demonstrated fact that
thermometry under this condition is impossible, and we are left without any
principle on which to found an absolute thermometric scale.
Next in importance to the primary establishment of an absolute scale,
independently of the properties of any particular kind of matter, is the fixing
upon an arbitrary system of thermometry, according to which results of
observations made by different experimenters, in various positions and
circumstances, may be exactly compared. This object is very fully attained by
means of thermometers constructed and graduated according to the clearly defined
methods adopted by the best instrument-makers of the present day, when the
rigorous experimental processes which have been indicated, especially by
Regnault, for interpreting their indications in a comparable way, are followed.
The particular kind of thermometer which is least liable to uncertain variations
of any kind is that founded on the expansion of air, and this is therefore
generally adopted as the standard for the comparison of thermometers of all
constructions. Hence the scale which is at present employed for estimating
temperature is that of the air-thermometer; and in accurate researches care is
always taken to reduce to this scale the indications of the instrument actually
used, whatever may be its specific construction and graduation.
The principle according to which the scale of the air-thermometer is graduated,
is simply that equal absolute expansions of the mass of air or gas in the
instrument, under a constant pressure, shall indicate equal differences of the
numbers on the scale; the length of a “degree” being determined by allowing a
given number for the interval between the freezing- and the boiling-points. Now
it is found by Regnault that various thermometers, constructed with air under
different pressures, or with different gases, give indications which coincide so
closely, that, unless certain gases, such as sulphurous acid, which approach the
physical condition of vapours at saturation, are made use of, the variations are
inappreciable[4]. This remarkable circumstance enhances very much the practical
value of the air thermometer; but still a rigorous standard can only be defined
by fixing upon a certain gas at a determinate pressure, as the thermometric
substance. Although we have thus a strict principle for constructing a definite
system for the estimation of temperature, yet as reference is essentially made
to a specific body as the standard thermometric substance, we cannot consider
that we have arrived at an absolute scale, and we can only regard, in
strictness, the scale actually adopted as an arbitrary series of numbered points
of reference sufficiently close for the requirements of practical thermometry.
In the present state of physical science, therefore, a question of extreme
interest arises: Is there any principle on which an absolute thermometric scale
can be founded? It appears to me that Carnot’s theory of the motive power of
heat enables us to give an affirmative answer.
The relation between motive power and heat, as established by Carnot, is such
that quantities of heat, and intervals of temperature, are involved as the sole
elements in the expression for the amount of mechanical effect to be obtained
through the agency of heat; and since we have, independently, a definite system
for the measurement of quantities of heat, we are thus furnished with a measure
for intervals according to which absolute differences of temperature may be
estimated. To make this intelligible, a few words in explanation of Carnot’s
theory must be given; but for a full account of this most valuable contribution
to physical science, the reader is referred to either of the works mentioned
above (the original treatise by Carnot, and Clapeyron’s paper on the same
subject).
In the present state of science no operation is known by which heat can be
absorbed, without either elevating the temperature of matter, or becoming latent
and producing some alteration in the physical condition of the body into which
it is absorbed; and the conversion of heat (or caloric) into mechanical effect
is probably impossible[5], certainly undiscovered. In actual engines for
obtaining mechanical effect through the agency of heat, we must consequently
look for the source of power, not in any absorption and conversion, but merely
in a transmission of heat. Now Carnot, starting from universally acknowledged
physical principles, demonstrates that it is by the letting down of heat from a
hot body to a cold body, through the medium of an engine (a steam-engine, or an
air-engine for instance), that mechanical effect is to be obtained; and
conversely, he proves that the same amount of heat may, by the expenditure of an
equal amount of labouring force, be raised from the cold to the hot body (the
engine being in this case worked backwards); just as mechanical effect may be
obtained by the descent of water let down by a water-wheel, and by spending
labouring force in turning the wheel backwards, or in working a pump, water may
be elevated to a higher level. The amount of mechanical effect to be obtained by
the transmission of a given quantity of heat, through the medium of any kind of
engine in which the economy is perfect, will depend, as Carnot demonstrates, not
on the specific nature of the substance employed as the medium of transmission
of heat in the engine, but solely on the interval between the temperature of the
two bodies between which the heat is transferred.
Carnot examines in detail the ideal construction of an air-engine and of a
steam-engine, in which, besides the condition of perfect economy being
satisfied, the machine is so arranged, that at the close of a complete operation
the substance (air in one case and water in the other) employed is restored to
precisely the same physical condition as at the commencement. He thus shews on
what elements, capable of experimental determination, either with reference to
air, or with reference to a liquid and its vapour, the absolute amount of
mechanical effect due to the transmission of a unit of heat from a hot body to a
cold body, through any given interval of the thermometric scale, may be
ascertained. In M. Clapeyron’s paper various experimental data, confessedly very
imperfect, are brought forward, and the amounts of mechanical effect due to a
unit of heat descending a degree of the air-thermometer, in various parts of the
scale, are calculated from them, according to Carnot’s expressions. The results
so obtained indicate very decidedly, that what we may with much propriety call
the value of a degree (estimated by the mechanical effect to be obtained from
the descent of a unit of heat through it) of the air-thermometer depends on the
part of the scale in which it is taken, being less for high than for low
temperatures.[6]
The characteristic property of the scale which I now propose is, that all
degrees have the same value; that is, that a unit of heat descending from a body
A at the temperature T░ of this scale, to a body B at the temperature (T-1)░,
would give out the same mechanical effect, whatever be the number T. This may
justly be termed an absolute scale, since its characteristic is quite
independent of the physical properties of any specific substance.
To compare this scale with that of the air-thermometer, the values (according to
the principle of estimation stated above) of degrees of the air-thermometer must
be known. Now an expression, obtained by Carnot from the consideration of his
ideal steam-engine, enables us to calculate these values, when the latent heat
of a given volume and the pressure of saturated vapour at any temperature are
experimentally determined. The determination of these elements is the principal
object of Regnault’s great work, already referred to, but at present his
researches are not complete. In the first part, which alone has been as yet
published, the latent heats of a given weight, and the pressures of saturated
vapour at all temperatures between 0░ and 230░ (Cent. of the air-thermometer),
have been ascertained; but it would be necessary in addition to know the
densities of saturated vapour at different temperatures, to enable us to
determine the latent heat of a given volume at any temperature. M. Regnault
announces his intention of instituting researches for this object; but till the
results are made known, we have no way of completing the data necessary for the
present problem, except by estimating the density of saturated vapour at any
temperature (the corresponding pressure being known by Regnault’s researches
already published) according to the approximate laws of compressibility and
expansion (the laws of Mariotte and Gay-Lussac, or Boyle and Dalton). Within the
limits of natural temperature in ordinary climates, the density of saturated
vapour is actually found by Regnault (╔tudes HydromΘtriques in the Annales de
Chimie) to verify very closely these laws; and we have reasons to believe from
experiments which have been made by Gay-Lussac and others, that as high as the
temperature 100░ there can be no considerable deviation; but our estimate of the
density of saturated vapour, founded on these laws, may be very erroneous at
such high temperatures at 230░. Hence a completely satisfactory calculation of
the proposed scale cannot be made till after the additional experimental data
shall have been obtained; but with the data which we actually possess, we may
make an approximate comparison of the new scale with that of the
air-thermometer, which at least between 0░ and 100░ will be tolerably
satisfactory.
The labour of performing the necessary calculations for effecting a comparison
of the proposed scale with that of the air-thermometer, between the limits of 0░
and 230░ of the latter, has been kindly undertaken by Mr. William Steele, lately
of Glasgow College, now of St. Peter’s College, Cambridge. His results in
tabulated forms were laid before the Society, with a diagram, in which the
comparison between the two scales is represented graphically. In the first
table, the amounts of mechanical effect due to the descent of a unit of heat
through the successive degrees of the air-thermometer are exhibited. The unit of
heat adopted is the quantity necessary to elevate the temperature of a
kilogramme of water from 0░ to 1░ of the air-thermometer; and the unit of
mechanical effect is a metre-kilogramme; that is, a kilogramme raised a metre
high.
In the second table, the temperatures according to the proposed scale, which
correspond to the different degrees of the air-thermometer from 0░ to 230░, are
exhibited. [The arbitrary points which coincide on the two scales are 0░ and
100░.]
Note.–If we add together the first hundred numbers given in the first table, we
find 135.7 for the amount of work due to a unit of heat descending from a body A
at 100░ to B at 0░. Now 79 such units of heat would, according to Dr. Black (his
result being very slightly corrected by Regnault), melt a kilogramme of ice.
Hence if the heat necessary to melt a pound of ice be now taken as unity, and if
a metre-pound be taken as the unit of mechanical effect, the amount of work to
be obtained by the descent of a unit of heat from 100░ to 0░ is 79x135.7, or
10,700 nearly. This is the same as 35,100 foot pounds, which is a little more
than the work of a one-horse-power engine (33,000 foot pounds) in a minute; and
consequently, if we had a steam-engine working with perfect economy at
one-horse-power, the boiler being at the temperature 100░, and the condenser
kept at 0░ by a constant supply of ice, rather less than a pound of ice would be
melted in a minute.
[1]Published in 1824 in a work entitled RΘflexions sur la Puissance Motrice du
Feu, by M. S. Carnot. Having never met with the original work, it is only
through a paper by M. Clapeyron, on the same subject, published in the Journal
de l’╔cole Polytechnique, Vol. XIV, 1834, and translated in the first volume of
Taylor’s Scientific Memoirs, that the Author has become acquainted with Carnot’s
Theory.–W.T. [note in Kelvin’s papers–CJG]
[2]An account of the first part of a series of researches undertaken by M.
Regnault by order of the French Government, for ascertaining the various
physical data of importance in the Theory of the Steam Engine, is just published
in the MΘmoires de l’Institut, of which it constitutes the twenty-first volume
(1847). The second part of the researches has not yet been published. [note in
Kelvin’s papers–CJG]
[3]A very important section of Regnault’s work is devoted to this object. [note
in Kelvin’s papers–CJG]
[4]Regnault, Relation des ExpΘriences, &c., Fourth Memoir, First Part. The
differences, it is remarked by Regnault, would be much more sensible if the
graduations were effected on the supposition that the coefficients of expansion
of the different gases are equal, instead of being founded on the principle laid
down in the text, according to which the freezing- and boiling-points are
experimentally determined for each thermometer. [note in Kelvin’s papers–CJG]
[5]This opinion seems to be nearly universally held by those who have written on
the subject. A contrary opinion however has been advocated by Mr Joule of
Manchester; some very remarkable discoveries which he has made with reference to
the generation of heat by the friction of fluids in motion, and some known
experiments with magneto-electric machines, seeming to indicate an actual
conversion of mechanical effect into caloric. No experiment however is adduced
in which the converse operation is exhibited; but it must be confessed that as
yet much is involved in mystery with reference to these fundamental questions of
natural philosophy. [note in Kelvin’s papers–CJG]
[6]This is what we might anticipate, when we reflect that infinite cold must
correspond to a finite number of degrees of the air-thermometer below zero;
since if we push the strict principle of graduation, stated above, sufficiently
far, we should arrive at a point corresponding to the volume of air being
reduced to nothing, which would be marked as -273░ of the scale (-100/.366, if
.366 be the coefficient of expansion); and therefore -273░ of the
air-thermometer is a point which cannot be reached at any finite temperature,
however low. [note in Kelvin’s papers–CJG]
>When you heat it to red hot, the electrons are going to higher energy levels and then emitting photons when they drop back down (I think).
Better to say that the charge in a solid isn’t evenly distributed, it’s concentrated in various spots according to molecular structure. The whole structure vibrates and wiggles around, by local distances proportional to the square root of absolute temperature and smaller than the atomic bonding distances (before the distances approximate the interatomic distances, the solid melts and vaporizes). Charge concentrated in spots that are little molecular groups can resonate, because the group is a little weight on a springy little attachment. Since the charge is wiggling around, it radiates electromagnetic energy (or, said more basically, as the charge moves, the electric field around it adapts, and the adaptations don’t happen everywhere at once, they propagate outward at speed c). Little resonating groups radiate much of their electromagnetic energy at the resonant frequency. Or resonant frequencies - they can be bending or twisting or pushing in and out, all at different resonant frequencies. This is, by the way, the basis of infrared spectroscopy.
I think the ‘temperature’ of many things (such as I believe the night sky) isn’t based at all on the classic thermodynamics definition of temperature, but instead on ‘black body’ radiation. The idea is that due to it’s heat, everything radiates electromagnetic waves (light, infrared, etc.), and that the object’s temperature determines how much and at what color (think about heating a piece of metal from room temperature to body heat, where it stands out on infrared but not regular light, to red hot to white hot). Every real object gives off heat radiation slightly differently (hence the infrared spectroscopy mentioned earlier), but an ideal featureless-even-on-an-atomic-level ‘black body’ follows a particular law.
So for the night sky, we determine it’s ‘temperature’ by looking at the radiation coming from it, and deciding what temperature an ideal black body would have to be to have the same kind of radiation. Turns out it’s 2.7K, so we say the temperature of the night sky is 2.7K.
And back to classical definitions of temperature, if we’re getting in to this level of detail, my rusty brain thinks the key quantity isn’t in fact ‘energy per degree of freedom’ but ‘marginal energy per degree of freedom’ or more precisely, ‘change in number of possible states per unit of energy added’. But we’re not teaching a semester-long thermodynamics class here, and it doesn’t really matter for thinking in general terms about things like how gasses heat up when compressed, or why water has a high heat capacity.