Understanding Heat and Its Role in Thermodynamics

[StarsRuler]

¿ What is heat, really, is the energy that in summatory of energies can´be written in function of temperature? It appears in primitive formulations of first principle of thermodynamics, but I don´t understand exactly its meaning

 

[berkeman]

¿ What is heat, really, is the energy that in summatory of energies can´be written in function of temperature? It appears in primitive formulations of first principle of thermodynamics, but I don´t understand exactly its meaning

There is a great Heat entry in the PF Library, but the Library appears broken at the moment. The Mentors and Admins are working on it...

 

[DrDu]

Heat is that part of change of internal energy of a thermodynamic system which is not due to classical work, i.e. mechanical or electrical or work done by electromagnetic fields.

 

[lightarrow]

¿ What is heat, really, is the energy that in summatory of energies can´be written in function of temperature? It appears in primitive formulations of first principle of thermodynamics, but I don´t understand exactly its meaning

Heat is the energy exchanged between two systems in contact, caused by a temperature difference only.
Note that it's incorrect to say "exchange of heat", even if many text write it.

 

[Andrew Mason]

¿ What is heat, really, is the energy that in summatory of energies can´be written in function of temperature? It appears in primitive formulations of first principle of thermodynamics, but I don´t understand exactly its meaning

"Heat" can be used in the colloquial sense to mean "thermal energy", which is the internal energy of a body due to its temperature (ie. the translational kinetic energies of molecules that follow the characteristic Maxwell-Boltzmann distribution).

But in thermodynamics, heat is used to mean Q (e.g. the first law: Q = ΔU + W) which is the flow of energy from one body to another due to a difference in temperature, as distinct from the energy flow due to mechanical work. This is how the concept of "heat" originated because it was thought that "heat" was a substance that flowed through bodies.

To avoid confusion between the colloquial use and the scientific use, Q is often referred to has "heat flow" rather than just "heat". This makes it clear that Q is not something that can exist in a body. Rather, it is an exchange of energy between bodies.

AM

 

[adjacent]

Heat is the measure of the average kinetic and potential energy of the molecules or atoms of a body.Heat is a type of energy-meaning it can be transformed from on form to another.Heat flows only from a higher temperature to a lower temperature.

 

[Andrew Mason]

Heat is the measure of the average kinetic and potential energy of the molecules or atoms of a body. Heat is a type of energy-meaning it can be transformed from on form to another. Heat flows only from a higher temperature to a lower temperature.

None of this is really correct:

1. Heat is the measure of the average kinetic and potential energy of the molecules or atoms of a body.

Internal energy (U), not heat (Q), is the measure of the average kinetic and potential energy of the molecules of a body in thermal equilibrium. (note: Temperature is the measure of the average translational kinetic energy of the molecules of a body in thermal equilibrium).

2. Heat is a type of energy-meaning it can be transformed from one form to another.

Heat (Q), or heat flow, represents a transfer of energy from one body to another. It is not correct to say that Heat is energy. Heat could be viewed as the work done at the molecular level by the molecules in one body on the molecules of another body.

Since heat is an energy transfer (molecular work) and not energy (energy being the ability to do work), heat cannot be converted into other forms of energy. Rather, internal energy of a body (eg. a thermal reservoir) can be converted into mechanical energy via heat (eg. heat flow that occurs in a heat engine).

3. Heat flows only from a higher temperature to a lower temperature.

This would be true if you were to add the word "spontaneously". As stated, it says that a refrigerator is not possible. Of course, refrigerators cause heat flow from cold to hotter bodies to occur by supplying mechanical work.

AM

 

[lightarrow]

I contest even the phrase "heat flow". If heat flows, what this flux is made of? Heat is just a transfer of energy (in peculiar conditions) so what "flows" is simply energy, not heat.

 

[Andrew Mason]

I contest even the phrase "heat flow". If heat flows, what this flux is made of? Heat is just a transfer of energy (in peculiar conditions) so what "flows" is simply energy, not heat.

The concept of "energy flow" or "energy transfer" is just mental bookkeeping. Energy is just a number. But since energy is always conserved we can think of energy moving around.

I agree that heat (Q), like work (W), is not something that a body possesses and it is not conserved. So it is not something that "flows" mathematically or physically. It is a particular kind of energy transfer process that occurs in a thermodynamic process. We could call it "a transfer of energy by means other than mechanical work", but heat or heat flow is the conventional term.

We often use terms in science that are well understood but not really correct, like "electromotive force". Although they are not really correct, they help us model the phenomena in our minds and have gained acceptance by usage.

AM

 

[tolove]

AM

Hey Andrew, could you check my understanding of heat? Is anything wrong with the following statement:

The internal energy (U) of an object (object being ideal single atom particles in either a gas, liquid, or solid state) is also referred to as the object's temperature (T) or translational kinetic energy (K).

If a relatively hot and cold substance are placed side by side (with a divider in the case of liquids and gasses to prevent convection, and neglecting radiation), the comparatively faster moving particles of the hot substance repeatedly slam into the slower particles belonging to the colder object. During this process, the temperature of the colder object rises, while the temperature falls in the hotter object. This process is described as a flow of heat (Q), even though no physical thing moves from one object to the other.

"Heat is the transfer of molecular work and not the ability to perform work."

I'm not sure I understand this quote.. could you elaborate on this? Heat (Q) is not conserved? If heat (Q) is defined in terms of changes in internal energy (U), and internal energy (U) is conserved, isn't heat flow (Q) also conserved?

Thank you very much for your time!

 

[Andrew Mason]

Hey Andrew, could you check my understanding of heat? Is anything wrong with the following statement:

The internal energy (U) of an object (object being ideal single atom particles in either a gas, liquid, or solid state) is also referred to as the object's temperature (T) or translational kinetic energy (K).

There is an important difference between internal energy and temperature. Temperature is a measure of one aspect of internal energy: the energy due to the motion of the centres of mass of the molecules that comprise the body. These molecules can have kinetic energy due to other kinds of motion - vibration and rotation about different axes - that do not involve the motion of the centres of mass of the molecules. These kinds of motion do not contribute to the body's temperature. Also, molecules can have potential energy due to inter-molecular forces. This potential energy is part of the internal energy but does not contribute to temperature.

If a relatively hot and cold substance are placed side by side (with a divider in the case of liquids and gasses to prevent convection, and neglecting radiation), the comparatively faster moving particles of the hot substance repeatedly slam into the slower particles belonging to the colder object. During this process, the temperature of the colder object rises, while the temperature falls in the hotter object. This process is described as a flow of heat (Q), even though no physical thing moves from one object to the other.

You have the right idea. But it is important to keep in mind that the speed distribution of molecules follows Maxwell-Boltzmann statistics: the molecules in the hotter body are not all moving faster than those in the cooler one but, statistically, the collisions result in the average kinetic energy of the molecules in the cooler body increasing and the average KE of the molecules in the hotter body decreasing. If it continues long enough, the two bodies reach thermal equilibrium and have the same temperature.

"Heat is the transfer of molecular work and not the ability to perform work."

I'm not sure I understand this quote.. could you elaborate on this?

Where did you get the quote? I said that heat could be viewed as the work done at the molecular level by the molecules in one body on the molecules of another body.

Work is something that a body does by applying a force through a distance. That work results in an "transfer" of energy ie a transfer of "the ability of a body to do work" to another body.

For example: A fast molecule colliding with a slow molecule. The fast molecule does work on the slow molecule, thereby increasing the slower molecule's kinetic energy. The slower molecule does negative work on the faster molecule (by applying an equal but opposite force through the same distance), thereby reducing the kinetic energy of the faster one. Since the increase in energy of the slow one is equal to the decrease in energy of the faster one, we can think of this as a "transfer of energy".

Heat (Q) is not conserved? If heat (Q) is defined in terms of changes in internal energy (U), and internal energy (U) is conserved, isn't heat flow (Q) also conserved?

Internal energy of a system is not conserved. It can change. It can change because energy can flow into our out of the system. It can flow into our out of the system because of heat flow into (Q), or out of (-Q), or work being done by (W), or on (-W), the system .

Heat and work represent transfers of energy into or out of the system. ΔU represents a change in energy of the system. The first law just says that the sum of all the transfers of energy into and out of the system must equal the change in energy of the system: ΔU = Q - W. Q, W and U can all change. But ΔU = Q - W is always true once everything settles down.

AM

 

[tolove]

AM

Could you see if I understood you correctly?

1) When neglecting subatomic properties, vibration, and rotation temperature (eg translational kinetic energy) is an exact measure of the internal energy of an 'ideal' substance.
2) Heat is an action. A substance cannot have heat (Q), rather, the substance has a temperature (T), which is a component of it's internal energy. When a temperature difference occurs, the substance performs a heat flow.
3) The internal energy of an isolated system is conserved. Components of the internal energy are not conserved, and may change form.

And an extra question on the subject: two perfectly insulated reservoirs of gas at equal temperatures are connected. In a modern view, is there a probability for a temperature difference to spontaneously occur between the two reservoirs?

Thank you very much again!

 

[Andrew Mason]

Could you see if I understood you correctly?

1) When neglecting subatomic properties, vibration, and rotation temperature (eg translational kinetic energy) is an exact measure of the internal energy of an 'ideal' substance.

You can say, for example, that for an ideal monatomic gas, temperature is proportional to the internal energy of the gas. That is because there are no inter-atomic forces and no rotational or vibrational modes in an ideal gas.

2) Heat is an action. A substance cannot have heat (Q), rather, the substance has a temperature (T), which is a component of it's internal energy. When a temperature difference occurs, the substance performs a heat flow.

Temperature is not a component of internal energy. It is a measure of one component of internal energy: translational KE.

Whether a temperature difference between two bodies leads to heat flow depends on the connection between them.

3) The internal energy of an isolated system is conserved. Components of the internal energy are not conserved, and may change form.

If the system is isolated, it cannot exchange energy (or mass) with its surroundings. So Q = 0 and W = 0. So what does the first law tell you about U (internal energy)?

What change in the components of internal energy would change? You will have to describe the system. If the system is a gas in an equilibrium state, then the "components" of the internal energy (rotational, translational, vibrational kinetic energy and potential energy) should not change. If the system was a weight, rope and pulley, and a piston in a cylinder filled with gas, the components of the system's energy may well change.

And an extra question on the subject: two perfectly insulated reservoirs of gas at equal temperatures are connected. In a modern view, is there a probability for a temperature difference to spontaneously occur between the two reservoirs?

You have to be careful in describing the reservoirs and the way they are connected. Are they at the same pressure? Is the volume fixed? Are they the same gas chemically? Does the connection allow the gases to mix?

AM

 

[tolove]

AM

Alright, I'm getting closer!

on 2), Translational KE is a component of internal energy; temperature is a measure of that motion.

on 3) For an isolated system, would it be better to say that internal energy remains constant rather than conserved? Those two words are very similar in meaning for me.

And the last part, I'm referring to the second law. In a classical sense, heat always spontaneously transfers from high to low temperature. However, in a modern view, is it correct to say that heat can spontaneously transfer from a low to a high temperature, just with an infinitesimally small chance of occurrence?

 

[Andrew Mason]

Alright, I'm getting closer!

on 2), Translational KE is a component of internal energy; temperature is a measure of that motion.

on 3) For an isolated system, would it be better to say that internal energy remains constant rather than conserved? Those two words are very similar in meaning for me.

ΔU=0

And the last part, I'm referring to the second law. In a classical sense, heat always spontaneously transfers from high to low temperature. However, in a modern view, is it correct to say that heat can spontaneously transfer from a low to a high temperature, just with an infinitesimally small chance of occurrence?

Even in a classical sense there is always a non-zero probability that a system will spontaneously move to a state of non-equilibrium. That is because the second law is a statistical law.

The second law would say that the probability of observing a measureable net transfer of energy from the cold to the hot for a measureable time interval is so small that it will never happen anywhere in a gazillion lifetimes of the universe. So we call it a law.

AM

 

[Rap]

Temperature is related to translational kinetic energy EXCEPT at extremely low temperatures, so its incorrect to state flatly that temperature is a measure of the average translational energy.

Temperature is simply related to the energy per degree of freedom when that degree of freedom is classical, i.e. the energy levels that are significantly populated are so close together that the distribution of energy is essentially over a continuum and the distribution is Boltzmann. Such a degree of freedom is termed "unfrozen". When quantum effects start to appear (energy level spacing not negligible with respect to average energy), then the degree of freedom is partially or completely frozen.

At room temperature, translational degrees of freedom are unfrozen, maybe some others (rotation, vibration, etc) are too, maybe not. Kinetic temperature equals thermodynamic temperature. As you lower the temperature, other degrees of freedom start to freeze. The last degrees of freedom to freeze are the translational degrees of freedom. When that happens, kinetic temperature diverges from thermodynamic temperature. This occurs at EXTREMELY low temperatures (e.g Bose Einstein condensation and non-Boltzmann distributions). Unless you are working in these regimes, kinetic temperature is a good proxy for thermodynamic temperature, but statements like "internal degrees of freedom do not contribute to temperature" are deeply misleading. Translational degrees of freedom are just like any other degree of freedom, except that they are the last to freeze as the temperature drops.

 

[Andrew Mason]

Temperature is related to translational kinetic energy EXCEPT at extremely low temperatures, so its incorrect to state flatly that temperature is a measure of the average translational energy.

That is the classical definition of temperature. See: http://hyperphysics.phy-astr.gsu.edu/hbase/kinetic/kintem.html

What is the definition of extremely low temperatures? At extremely low temperatures, the temperature of molecules is determined by directly measuring the translational speed of the molecules. That is how they measure temperatures in nano-Kelvins.

How are you suggesting temperature should be defined?

Temperature is simply related to the energy per degree of freedom when that degree of freedom is classical, i.e. the energy levels that are significantly populated are so close together that the distribution of energy is essentially over a continuum and the distribution is Boltzmann. Such a degree of freedom is termed "unfrozen". When quantum effects start to appear (energy level spacing not negligible with respect to average energy), then the degree of freedom is partially or completely frozen.

But in cases where the other degrees of freedom are active, the temperature is still determined the average translational kinetic energy of the molecules.

The problem with defining temperature as the average energy per degree of freedom is that it assumes that the equipartition theorem always applies. And, of course, it does not apply in many situations - not just at low temperatures.

Unless you are working in these regimes, kinetic temperature is a good proxy for thermodynamic temperature, but statements like "internal degrees of freedom do not contribute to temperature" are deeply misleading. Translational degrees of freedom are just like any other degree of freedom, except that they are the last to freeze as the temperature drops.

Internal degrees of freedom contribute to heat capacity. So they certainly will affect the temperature of a body experiencing heat transfer. But the measure of that body's temperature is still determined by the average translational kinetic energy of the molecules.

Can you give us an example of any substance in which the translational degrees of freedom are frozen at some temperature?

AM

 

[Rap]

That is the classical definition of temperature. See: http://hyperphysics.phy-astr.gsu.edu/hbase/kinetic/kintem.html

What is the definition of extremely low temperatures? At extremely low temperatures, the temperature of molecules is determined by directly measuring the translational speed of the molecules. That is how they measure temperatures in nano-Kelvins.
How are you suggesting temperature should be defined?AM

Thermodynamic temperature is defined in the second law of thermodynamics, usually using a Carnot cycle. It makes no reference to particles or statistical mechanics. The second law says that the heat transferred \delta Q can be expressed as T\,dS where the thermodynamic temperature T is an intensive state function and entropy S is an extensive state function. It leaves the scales hanging, pick your own, Kelvin, Rankine, whatever.

But in cases where the other degrees of freedom are active, the temperature is still determined the average translational kinetic energy of the molecules.AM

No, in this case, you can just as well say the temperature is determined by any, some or all of the internal degrees of freedom (DOFs). Suppose you have five internal degrees of freedom, 3 translational, 2 rotational. As long as all those DOFs are unfrozen and sharing energy, you can just as well say that temperature is determined by the average total energy of the molecules, E=(5/2)kT, or by the translational DOFs: E=(3/2)kT, or by the internal DOFs: E=(2/2)kT or by any single DOF: E=(1/2)kT. Why not say that the x-component of the speed determines the temperature? You could, because all three translational DOFs are unfrozen and sharing energy, so it wouldn't matter if you did. You get the same answer no matter which degree(s) of freedom you choose.

The problem with defining temperature as the average energy per degree of freedom is that it assumes that the equipartition theorem always applies. And, of course, it does not apply in many situations - not just at low temperatures.AM

Whenever it does not apply to a particular DOF, that DOF is partially or (practically) completely frozen. Every DOF has a temperature below which it must be described quantum mechanically, and the equipartition theorem becomes invalid. It begins to freeze. Below that temperature is "low temperature" for that DOF. Low temperatures for rotational DOFs in common gases may be room temperature. For translational DOF's "low temperature" is down in the nanokelvin range. That's what makes translational DOFs special - they are the last to freeze, but freeze they will.

Can you give us an example of any substance in which the translational degrees of freedom are frozen at some temperature?AM

A gas of bosons in a box, in which almost all the particles are at the ground energy level, only a small percentage in the first translational excited state, even less in the higher excited states. In other words, a Bose-Einstein condensate. Translational DOF's are nearly completely frozen. (No DOF is ever completely frozen except at unattainable absolute zero)

 

[Andrew Mason]

Thermodynamic temperature is defined in the second law of thermodynamics, usually using a Carnot cycle. It makes no reference to particles or statistical mechanics. The second law says that the heat transferred \delta Q can be expressed as T\,dS where the thermodynamic temperature T is an intensive state function and entropy S is an extensive state function. It leaves the scales hanging, pick your own, Kelvin, Rankine, whatever.

Do you really want to use a definition of temperature that pre-dates kinetic theory?

No, in this case, you can just as well say the temperature is determined by any, some or all of the internal degrees of freedom (DOFs). Suppose you have five internal degrees of freedom, 3 translational, 2 rotational. As long as all those DOFs are unfrozen and sharing energy, you can just as well say that temperature is determined by the average total energy of the molecules, E=(5/2)kT, or by the translational DOFs: E=(3/2)kT, or by the internal DOFs: E=(2/2)kT or by any single DOF: E=(1/2)kT. Why not say that the x-component of the speed determines the temperature? You could, because all three translational DOFs are unfrozen and sharing energy, so it wouldn't matter if you did. You get the same answer no matter which degree(s) of freedom you choose.

IF the equipartition theorem actually applied perfectly, of course you could say that the temperature is a measure of the kinetic energy associated with any of the active degrees of freedom - because they are all equal. But the problem is that it often does not apply. So what then do you use to determine the temperature? Answer: the translational kinetic energy. There is good reason for this: it is the difference in the translational KE between two bodies that causes spontaneous heat flow.

A gas of bosons in a box, in which almost all the particles are at the ground energy level, only a small percentage in the first translational excited state, even less in the higher excited states. In other words, a Bose-Einstein condensate. Translational DOF's are nearly completely frozen. (No DOF is ever completely frozen except at unattainable absolute zero)

And how do they measure/define the temperature of a Bose-Einstein condensate?

AM

 

[atyy]

That is the classical definition of temperature. See: http://hyperphysics.phy-astr.gsu.edu/hbase/kinetic/kintem.html

That article seems to support Rap's second law based definition, since it claims to define "kinetic temperature", not "temperature".

The same site has a discussion http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/temper2.html#c2 , and says at http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/temper2.html#c1 "Temperature is expressed as the inverse of the rate of change of entropy with internal energy, with volume V and number of particles N held constant. This is certainly not as intuitive as molecular kinetic energy, but in thermodynamic applications it is more reliable and more general."

 

[Rap]

Do you really want to use a definition of temperature that pre-dates kinetic theory?AM

Yes - Kinetic theory and statistical mechanics explain classical thermodynamics, they do not replace it. Temperature is not defined in kinetic theory or statistical mechanics, it is defined in classical thermodynamics, and a microscopic understanding (but not definition) of temperature is provided by kinetic theory and statistical mechanics. Same for entropy, internal energy, chemical potential, and the relationship between Avogadro's number and the mole.

IF the equipartition theorem actually applied perfectly, of course you could say that the temperature is a measure of the kinetic energy associated with any of the active degrees of freedom - because they are all equal. But the problem is that it often does not apply. So what then do you use to determine the temperature? Answer: the translational kinetic energy. There is good reason for this: it is the difference in the translational KE between two bodies that causes spontaneous heat flow.

That's probably mostly true, but not true in principle. If you have two bodies, both with, say, rotational and translational DOFs, all the DOFs are sharing energy. Translational energy is converted to rotational energy, and vice versa, by collisions between both like particles and unlike particles. If the DOFs are classical (unfrozen), then equipartition says all DOFs wind up having the same average energy: kT/2. What if one body (A) has its translational DOFs and rotational DOFs at different temperatures, while both bodies' (A and B) translational DOFs are at the same temperature? My first impulse, and probably yours, would be to say that the unequal temperatures in A would equilibrate quickly, and then the two bodies would be at different temperatures, and heat would flow via the translational DOFs. But what if the equilibration of the two temperatures in A were slow compared to the rate at which heat flowed from the translational DOFS of B to the rotational DOFs of A? Then heat would mostly flow from the translational DOFs of B to the rotational DOFs of A. I only come up with this scenario to illustrate a principle, I will be really scratching my head to find a concrete example.

And how do they measure/define the temperature of a Bose-Einstein condensate?

Well, that's a good question. A good article on the theory of Bose-Einstein condensates is at www.stanford.edu/~rsasaki/AP389/AP389_chap3.pdf

Just after Eq. 3.37 it states:

"That is, the system volume must be much larger than the cube of thermal de Broglie
wavelength. The use of the inequality (...) is actually crucial in the above
theory of BEC based on the energy density of states and the continuous energy integral
rather than discrete sum over the single particle excited states."

So I was wrong to say a BEC is an example of translational DOFs being frozen. Translational DOFs are frozen when the thermal de Broglie wavelength cubed is of the order of the volume of the system, for which the usual analysis of BEC (as given in the above source) is not correct. So I guess the temperature of a BEC is usually measured using the fact that the translational DOFs are in a state of Bose-Einstein (i.e. continuum) equilibrium and equipartition holds. In other words, by the kinetic temperature.

The thermal de Broglie wavelength (TdBW) is the quantum wavelength of a particle as a function of temperature. When the temperature is so low that the TdBW is of the order of a spatial dimension of the system, the spacing between the translational energy levels is of the same order as the translational energy itself. When that happens, you don't have equipartition, the translational DOFs are freezing up, you must use a "discrete sum over the single particle excited states" (to quote the above source) and you cannot use the translational energy as a measure of temperature.

So I've ejected a misconception of mine due to this discussion, this is good. But what is the temperature at which the TdBW is of the order of the system size? The TdBW is given by: \Lambda=\frac{h}{2\pi m k T} and choosing a particle mass of, say, 1 AMU, I calculate that the TbDW to be one centimeter at a temperature of 3 \times 10^{-14} K

So you can easily make the point that the kinetic temperature (temperature derived from translational DOFs) is equal to the thermodynamic temperature for any macroscopic system of practical interest, and I can still make the point that the kinetic temperature and the thermodynamic temperature are not identical.

 

[atyy]

I can still make the point that the kinetic temperature and the thermodynamic temperature are not identical.

Were you thinking of something like this?

http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/temper2.html#c1

"The concept of temperature is complicated by internal degrees of freedom like molecular rotation and vibration and by the existence of internal interactions in solid materials which can include collective modes. The internal motions of molecules affect the specific heats of gases, with diatomic hydrogen being the classic case. Collective modes affect the specific heats of solids, particularly at low temperatures."

 

[tolove]

Just wanted to say I'm still following this thread. Thank you guys for the in depth discussion, it is very helpful.

 

[Andrew Mason]

That article seems to support Rap's second law based definition, since it claims to define "kinetic temperature", not "temperature".

The same site has a discussion http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/temper2.html#c2 , and says at http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/temper2.html#c1 "Temperature is expressed as the inverse of the rate of change of entropy with internal energy, with volume V and number of particles N held constant. This is certainly not as intuitive as molecular kinetic energy, but in thermodynamic applications it is more reliable and more general."

There are different ways to define temperature. But they must all be equivalent. Ultimately, how ever it is defined, a temperature difference between two bodies in thermal contact with each other can only result in a spontaneous heat flow from the body with the higher temperature to the one with the lower temperature.

Defining T = 1/(dS/dU)_N,V is equivalent to kinetic temperature. It may be more general but it involves concepts that are difficult for students to understand. It also does not, it seems to me, explain much. It is not easy to see why, physically, spontaneous heat flow occurs only from hot to less hot.

Defining T = dQ/dS is not correct. T = dQ_rev/dS would be ok, but since dQ_rev = TdS = dU + PdV, this is just: T = 1/(dS/dU)_N,V with V, N constant

Except for a few circumstances where it may not be possible to measure translational speeds, saying that the temperature of a body is proportional to the average translational kinetic energy of its constituent molecules provides an accurate and understandable definition.

AM

 

[Andrew Mason]

What if one body (A) has its translational DOFs and rotational DOFs at different temperatures, while both bodies' (A and B) translational DOFs are at the same temperature?

If equipartition applies, how does A end up with the translational and rotational DOF's at different temperatures?

Well, that's a good question. A good article on the theory of Bose-Einstein condensates is at www.stanford.edu/~rsasaki/AP389/AP389_chap3.pdf

An even better question would be: does temperature of Bose-Einstein condensate have any meaning, since the atoms in BEC follow Bose, not Maxwell-Boltzmann, statistics?

AM

 

[Khashishi]

Heat is one of those terms whose exact definition depends on the context and how fundamental a picture you are using.

 

[Rap]

If equipartition applies, how does A end up with the translational and rotational DOF's at different temperatures? AM

It was in answer to your statement "it is the difference in the translational KE between two bodies that causes spontaneous heat flow". If equipartition applies, how do the two bodies end up with different translational KE's? What I'm saying is that you don't need two distinct bodies to have temperature differences. In a plasma in an electric field, the electrons will be accelerated more than the heavy particles. They will have a different temperature than the heavy particles. Likewise you can have different DOFs at different temperatures, and there will be an equilibration process that occurs. Suppose you had a gas in equilibrium, with 3 translational, 2 rotational DOFs, and the rotational DOFs equilibrated to the translational DOFs with some time constant tr, much longer than the translational DOFs equilibrated with each other (tt). If you could manage to chill the gas in a time interval much shorter than tr, but longer that tt, the rotational DOFs would be out of equilibrium with the translational DOFs: i.e. they would have a different temperature.

An even better question would be: does temperature of Bose-Einstein condensate have any meaning, since the atoms in BEC follow Bose, not Maxwell-Boltzmann, statistics?AM

Temperature is defined by the second law, not by Maxwell Boltzmann statistics. Maxwell Boltzmann statistics explains how the classically defined thermodynamic temperature relates to the energy distribution of particles which sparsely populate the energy levels, so sparsely that you can assume only 1 particle per energy level. When the energy levels are not sparsely populated, you have two cases - Bosons and Fermions, and the statistics are different. Bose-Einstein statistics explains how thermodynamic temperature relates to the energy distribution of Bosons, Fermi-Dirac statistics explains how thermodynamic temperature relates to the energy distribution of Fermions. Sparse or not. When the population of the energy levels are sparse, the plus or minus 1 in the denominator of the BE or FD distributions is negligible, and you are left with a Maxwell Boltzmann distribution. The thermodynamic temperature in all the equations is defined by the second law.

I agree, the second law definition is more difficult than the kinetic temperature concept, but its the real definition, the kinetic temperature is not. The reason its more difficult is because in the second law, temperature and entropy are inextricably linked. You cannot really understand temperature until you understand entropy and vice versa, and entropy is a real bear to understand. I still don't fully get it. I understand the statistical mechanics explanation of entropy, the information theory of entropy, but, as you say, the thermodynamic definition of temperature is usually given in terms of entropy, which is not very intuitively useful, since temperature and entropy are so inextricably linked. In classical thermodynamics, the temperature is more accessible to understanding, while the entropy is hard to understand. In statistical mechanics, it is the entropy that is easier to understand, the temperature is more difficult. Saying the temperature is the kinetic temperature is the easy way out, and intuitively helpful, but its just not identically correct. Caratheodory and Lieb & Yngvason are people who have struggled with the classical understanding of temperature/entropy and perhaps I will fully understand it some day. See http://books.google.com/books/about/The_Entropy_Principle.html?id=kxOXGfNpJKEC for Thess's description of the Lieb & Yngvason approach. Its kind of like Caratheodory's, but better, I think. A statistical mechanical theory, when applied to classical thermodynamics, makes no predictions other than to agree with classical thermodynamics. If it fails to do so, then the SM theory is wrong, not classical thermodynamics. I don't see how one can fully appreciate and understand the SM concepts of temperature and entropy unless one understands the classical thermodynamic concepts, where they are actually defined, rather than just explained as in statistical mechanics and kinetic theory.

 

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The concept of "energy flow" or "energy transfer" is just mental bookkeeping. Energy is just a number. But since energy is always conserved we can think of energy moving around.

I agree that heat (Q), like work (W), is not something that a body possesses and it is not conserved. So it is not something that "flows" mathematically or physically. It is a particular kind of energy transfer process that occurs in a thermodynamic process. We could call it "a transfer of energy by means other than mechanical work", but heat or heat flow is the conventional term.

I know, I have used it for decades. But what about suggesting books authors to think a bit about it?
If a body A heats a body B sending it a laser beam, is it transferring heat?

It doesn't, because...

 

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About kinetic temperature there probably (as far as I know) are also other problems: to define radiation temperature; to define the temperature of a few molecules only.