[Bored chemist]

It can't be a very good attempt; it shows the solid as full of springs, and then it ignores potential energy.

Perhaps the potential energy isn't counted as (doesn't contribute to) temperature.

It is counted and does contribute.
Why choose to be wrong about that?

Then the definition would be false.

What's to explain? The caption statement is almost correct:  the definition of temperature is the average internal kinetic energy of a body. You can't explain a definition!

Alan is wrong. That's not news.

[hamdani yusuf (OP)]

What's to explain? The caption statement is almost correct:  the definition of temperature is the average internal kinetic energy of a body. You can't explain a definition!

How do you distinguish between internal and external energy?

What do you think about a resonating tuning fork?
What do you think about a resonating LC circuit?
Electrical current circulating indefinitely in a superconducting ring?

[hamdani yusuf (OP)]

It is counted and does contribute.

As an example, an object absorbs 2 Joule of energy. 1 Joule is converted to potential energy, and 1 Joule is converted to kinetic energy.
Another object with same mass absorbs 2 Joule of energy. 0 Joule is converted to potential energy, and 2 Joules is converted to kinetic energy.
According to the definition above, the temperature of the object increases corresponding to the increase of kinetic energy. Hence the second object increases its temperature twice as much as the first object.

[Eternal Student]

Hi.

...an object absorbs 2 Joule of energy. 1 Joule is converted to potential energy, and 1 Joule is converted to kinetic energy.  Another object with same mass absorbs 2 Joule of energy. 0 Joule is converted to potential energy, and 2 Joules is converted to kinetic energy....... the second object increases its temperature twice as much as the first object... 

   Not quite.  In the general case, one of the following applies:

(i)  The second object is not yet in thermal equillibrium.   There isn't a equal partition of energy between the various modes or degrees of freedom it can support as forms of internal energy.  As such its temperature is not well defined yet.

(ii)  The "potential energy" you were considering was never one of the ways in which it can support internal energy,  for example it might be gravitational potential energy due to lifting the object up higher.   It had no relevance for temperature.  (In which case, you would be right,  gaining that sort of potential energy didn't make the first object hotter, it was 1 J of energy in some form that didn't change its temperature).

Best Wishes.

[hamdani yusuf (OP)]

(i)  The second object is not yet in thermal equillibrium.   There isn't a equal partition of energy between the various modes or degrees of freedom it can support as forms of internal energy.  As such its temperature is not well defined yet.

Then wait until it is well defined.

(ii)  The "potential energy" you were considering was never one of the ways in which it can support internal energy,  for example it might be gravitational potential energy due to lifting the object up higher.   It had no relevance for temperature.  (In which case, you would be right,  gaining that sort of potential energy didn't make the first object hotter, it was 1 J of energy in some form that didn't change its temperature).

If we add thermal energy to an object but its temperature doesn't change, then according to the definition above, its internal kinetic energy doesn't change. Hence, the energy should be converted into something else, which can be external kinetic energy, external potential energy, internal potential energy, or combinations of those kinds of energy.
Melting ice may cross our minds as an example.

[Eternal Student]

Hi.

Then wait until it is well defined.

    The fundamental idea is that the energy will automatically be re-distributed throughout the system, so that you have an equi-partition of energy in all the modes or degrees of freedom that the system can support.    So, if you do wait, then some of the energy,  the 2 J of internal kinetic energy will be passed to other modes of storing energy.   Taking a generalised example, if your system supports kinetic energy of the particles and also potential energy in the bonds (or springs) between atoms  then you end up with 1 J remaining as kinetic   and  1 J being potential energy.       i.e.   The second object becomes exactly like the first object in your example.

If we add thermal energy to an object but its temperature doesn't change, then according to the definition above, its internal kinetic energy doesn't change. Hence, the energy should be converted into something else, which can be external kinetic energy, external potential energy, internal potential energy, or combinations of those kinds of energy.
Melting ice may cross our minds as an example.

    As mentioned before, Temperature is the measure of energy in ANY one degree of freedom or mode of supporting internal energy that the system has.   Vector geometry is on our side, so the average overall translational kinetic energy, ½mv²,  of molecules (in something like a simple gas) can also be considered as something proportional to temperature.

             [By vector geometry,  v_x = x component of v etc.]

          [Bar over = average over all particles.  Note that it is the mean square not the square of the mean.]

But due to equi-partition,     =  the (average) energy of a particle in any one mode  =    . (Total Energy in one mode)    (where N = number of particles in the substance).

So the average overall translational k.e.  =   3 times  average k.e. in one mode  =  something proportional to Temperature.   Both the overall translational k.e.  AND  the energy in any one mode are proportional to temperature, just with different constants of proportionality.   In simple kinetic theory of gases we have:
     (Total Energy in any one mode) = ½ K_B . T     with  T = temperature,  K_B = Boltzmann constant.

   Now we can consider what might be happening when there is a change of state:
   Sometimes when there is a phase change  (let's say solid to gas),  the new system (the gas) supports more modes of storing energy.   As a simple example consider a diatomic gas.   While in the gas state, it can support 2 different rotations that were not possible while the molecules were bound in the solid state.    Temperature is proportional to the energy in ANY one degree of freedom you choose to use.     
    So one thing that can happen when there is a change of state is that as parts of the system break off from the solid state and form the next state, there simply are more modes of storing energy.   So the energy in any ONE mode can remain constant (the temperature doesn't change) but the total internal energy stored in the substance does increase.   So in answering "where does the latent heat go?",  we can say that it is stored internally in the substance, it doesn't result in any increase in the translational k.e. of the particles (and hence no change in temperature) because there is a whole new mode of internal energy now available and energy has to be re-distributed to that new mode ("fill it up") to maintain an equi-partition of energy for the system.

    The new system of defining temperature (since May 2019) bases everything on kinetic theory.   It's very new and not something I'm very familiar with.  There does appear to be a change in the number of modes for supporting internal energy when a substance changes state but the discussion above is a simplification of the situation.

References:
    https://en.wikipedia.org/wiki/Temperature#Kinetic_theory_approach
(and more generally the entire article about Temperature, where the very new adjustments as of May 2019 are still only just being introduced, edited and corrected even now).

Best Wishes.

[alancalverd]

How do you distinguish between internal and external energy?

It is the average kinetic energy of the particles inside the body, regardless of how fast the body is moving or what its gravitational potential may be.

[alancalverd]

As an example, an object absorbs 2 Joule of energy. 1 Joule is converted to potential energy, and 1 Joule is converted to kinetic energy.
Another object with same mass absorbs 2 Joule of energy. 0 Joule is converted to potential energy, and 2 Joules is converted to kinetic energy.
According to the definition above, the temperature of the object increases corresponding to the increase of kinetic energy. Hence the second object increases its temperature twice as much as the first object.

Correct.

An increase of internal potential energy would correspond to a partial or total change of state within the body.

I encountered this when measuring radiation dose with a calorimeter. Dose is defined as energy aborbed per unit mass, and the principal concern for radiation protection and radiotherapy is the measurement of dose to water. For practical simplicity most primary standard calorimeters use graphite as the absorber because it is mechanically stable and has about a tenth of the specific heat capacity of water so undergoes a larger temperature change (a lethal dose of ionising radiation raises your body temperature by about 0.001 degree - my task was to measure that to ± 10^-6K). One of my colleagues built a water calorimeter - rather less portable device but clearly worth directly measuring the quantity of interest rather than trying to derive it. Problem was that the water calorimeter generally measured about 3% less than the graphite calorimeter, though both were calibrated to  ± 0.01%. I thought the difference was due to "virgin" water forming metastable polymers when irradiated, because the defect gradually decreased with extended irradiation to high doses but later work has revealed all sorts of complex chemistry possible with just H and O atoms and plenty of energetic photons.

[Bored chemist]

As an example, an object absorbs 2 Joule of energy. 1 Joule is converted to potential energy, and 1 Joule is converted to kinetic energy.
Another object with same mass absorbs 2 Joule of energy. 0 Joule is converted to potential energy, and 2 Joules is converted to kinetic energy.
According to the definition above, the temperature of the object increases corresponding to the increase of kinetic energy. Hence the second object increases its temperature twice as much as the first object.

Correct.

An increase of internal potential energy would correspond to a partial or total change of state within the body.

I encountered this when measuring radiation dose with a calorimeter. Dose is defined as energy aborbed per unit mass, and the principal concern for radiation protection and radiotherapy is the measurement of dose to water. For practical simplicity most primary standard calorimeters use graphite as the absorber because it is mechanically stable and has about a tenth of the specific heat capacity of water so undergoes a larger temperature change (a lethal dose of ionising radiation raises your body temperature by about 0.001 degree - my task was to measure that to ± 10^-6K). One of my colleagues built a water calorimeter - rather less portable device but clearly worth directly measuring the quantity of interest rather than trying to derive it. Problem was that the water calorimeter generally measured about 3% less than the graphite calorimeter, though both were calibrated to  ± 0.01%. I thought the difference was due to "virgin" water forming metastable polymers when irradiated, because the defect gradually decreased with extended irradiation to high doses but later work has revealed all sorts of complex chemistry possible with just H and O atoms and plenty of energetic photons.

You should have asked a chemist.
We know that irradiating water makes things like hydrogen peroxide and we know that making rocket fuel takes energy.

[alancalverd]

It was pretty obvious that some chemistry was going on because the mechanism for radiation damage in living cells involves the production of free radicals in the cytoplasm (mostly water) that disrupt DNA. The surprise was the persistence of these reaction products in pure water - we'd expected everything to recombine in microseconds. It is remarkable stuff!

[Bored chemist]

- we'd expected everything to recombine in microseconds.

You do know that you can just buy hydrogen peroxide as a solution in water, don't you?

[hamdani yusuf (OP)]

The fundamental idea is that the energy will automatically be re-distributed throughout the system, so that you have an equi-partition of energy in all the modes or degrees of freedom that the system can support.    So, if you do wait, then some of the energy,  the 2 J of internal kinetic energy will be passed to other modes of storing energy.   Taking a generalised example, if your system supports kinetic energy of the particles and also potential energy in the bonds (or springs) between atoms  then you end up with 1 J remaining as kinetic   and  1 J being potential energy.       i.e.   The second object becomes exactly like the first object in your example.

Let's put your analysis to a concrete example. The first object is a diatomic gas, such as N2, while the second object is a monoatomic gas, such as Neon. Is it still valid?

[Bored chemist]

OK, lets consider them.
Take the same number of molecules of each gas starting from very cold, and add heat slowly.
In order to get cold enough, you might need to work at low pressure to stop them turning to liquid or solid,but that's a different issue.
And we can use a large box, to address a different issue, but I will come back to that.
At low enough temperatures, the heat capacities of the gases are the same- the heat that you add goes into making the molecules translate.
So rises in the temperatures of the two gases are both the same.

But there comes a point (A few K, I think) where there is enough energy to make the N2 molecules rotate.
(Not many people think about this, but rotational energy is quantised).
So the heat that you add to the neon all goes into increasing the translational energy of the molecules.
But some of the energy you add to the nitrogen goes into making the nitrogen molecules rotate.

So now you have the same amount of heat added to both gases, but in the case of the nitrogen it is shared out amend 5 degrees of freedom, but with the neon it is only split between 3 degrees of freedom.

So the temperature rise- the average energy per degree of freedom- is smaller for the nitrogen.

For a given amount of heat, the neon gets hotter than the nitrogen.

Eventually, you get the gas hot enough that you start to excite the vibration of the nitrogen molecule as well, and its heat capacity rises again.
And then as you continue to heat both gases, the rates of change of temperature become even more different.

So, it  is simply not true to say you can ignore everything but translational KE.

For what it's worth, the translational KE is also quantise and, for an atom in a small enough box, the temperature is rather poorly defined but that's seldom a practical issue.

The important thing is that what the video says is simply wrong.

[Eternal Student]

Hi.

   I don't have much disagreement with what @Bored chemist  has said.    That's a very detailed examination of what might happen as you start to heat  N₂.    In particular, experimental measurements of the specific heat capacity does show slight variation with temperature (at least old definition temperature).

    The only issue I would have is that is that I don't think we should use any Quantum Mechanics or attempt to quantise everything.   Temperature (modern temperature) does not concern itself with a precise description of anything, it's just a useful concept in statistical mechanics which relies very much on classical mechanics being applied to the microscopic particles instead of quantum mechanics.

   .....Since May, 2019, its degrees (they meant degrees Kelvin) have been defined through particle kinetic theory, and statistical mechanics....

[From Wikipedia:  under sub-category "International Kelvin Scale" of this page https://en.wikipedia.org/wiki/Temperature]

     .....Kinetic theory provides a microscopic account of temperature for some bodies of material, especially gases, based on macroscopic systems' being composed of many microscopic particles, such as molecules and ions of various species, the particles of a species being all alike. It explains macroscopic phenomena through the classical mechanics of the microscopic particles.

[Taken from the  "Kinetic theory approach" sub-section of the above article in Wikipedia.]

    What does this mean in practical terms?    Rotational k.e. should, in reality be quantised, as discussed by @Bored chemist ,  however the very latest (since 2019) definition of temperature for the N₂ simply doesn't care.     

   I'm not responsible - I didn't change the way temperature is defined or thought about.   Personally, I think the decision to use classical mechanics in their models has seriously limited the expected lifetime of this new approach to defining temperature.   We'll have a new approach in 20 years, which probably will start using Quantum Mechanics or else return to using a Thermodynamic temperature definition.

Best Wishes.

[hamdani yusuf (OP)]

At low enough temperatures, the heat capacities of the gases are the same- the heat that you add goes into making the molecules translate.
So rises in the temperatures of the two gases are both the same.

Why don't polyatomic gases rotate nor vibrate at low temperature?

[hamdani yusuf (OP)]

But there comes a point (A few K, I think) where there is enough energy to make the N2 molecules rotate.
(Not many people think about this, but rotational energy is quantised).

What's the minimum non-zero quantity of rotational energy?
Is vibrational energy also quantized?

[Eternal Student]

Hi.

Why don't polyatomic gases rotate nor vibrate at low temperature?

    In the models they do rotate slowly at any temperature above 0 Kelvin.   QM is not important.   There would be a continuous probability distribution function to describe the probability of any particle having a given rotation.

    In reality they may or may not because angular momentum should be quantised according to QM.   If you had a large number of N₂ molecules you may find a discrete distribution for their rotations.   Some have 0 angular momentum,  others should have angular momentum =  n . ħ/2   for some positive integer n.

Best Wishes.

[Bored chemist]

Why don't polyatomic gases rotate nor vibrate at low temperature?

Because the energy available to them is less than that required to get them  to rotate or vibrate.

Is vibrational energy also quantized?

Yes.

QM is not important. 

Yes it is.

What's the minimum non-zero quantity of of rotational energy?

It depends on the molecule.
For N2 I think it's about 10^-4 eV.

[Eternal Student]

Hi.

QM is important in reality.   That's not in dispute.

However, it does not appear in the classical microscopic mechanical models that are used to define temperature since May 2019.
    Temperature (in kelvin, since 2019)  is  NOT  telling you something about the actual kinetic energy of a particle you can observe in reality.   It is much more abstract than that,  it is the kinetic energy a particle would have in the model.
    Let's not lose sight of the idea that particles might not even exist in reality, everything might be waves or some such thing.    However, there are always particles in the model,  they have kinetic energy and classical mechanics is used.

     This is actually a fairly small and unimportant issue and it's hardly worth distracting @hamdani yusuf  from whatever the main topic was originally hoping to cover.   However, it's interesting, i.m.o.  so I'll spend another moment discussing it.
      In practice it's not like the shift in considering what temperature is supposed to be maters a lot.   At temperatures above a few degrees kelvin and for a large number of particles you are out of the regime of  Q.M.   and there is good agreement between a classical microscopic model and properties you might actually observe in reality.   However, just for the sake of completion, it must be noted that "the temperature" of a body, under the latest definitions and international agreements for the kelvin scale has an entirely theoretical basis....
      There is a paradigm shift in what "temperature" is supposed to be.   It is not a description of some real physical property like kinetic energy that particles actually have in reality.  Instead it is a description of the average kinetic energy that a particle would have in the model due to the energy that has been transferred to it.   The numerical value of the temperature in kelvin (since May 2019) is only telling you something that is easily identified in the model and may be quite impossible to measure or observe in reality.   It just turns out that for a macroscopic object (and above extremely low energies like a few degrees kelvin) classical microscopic mechanics does describe the situation well and a property emerges in the macroscopic object that behaves as we would hope and has the properties we would want to describe as the "temperature" of that real life macroscopic body.
     As outlined earlier, for right or wrong, the scientists have used classical mechanics and not QM in constructing their models for the microscopic thermal behaviour modelling of substances.   That's not my fault, that's just what they've done.

    --------------

Returning to the original question of the OP:
   What is temperature?

   Reasonable answers and discussions have been made in countless earlier posts.   It might be best to say there are now several different approaches to defining temperature.
   Wikipedia describes some of these approaches ....  empirical,  theoretical,  thermodynamic,  ... etc.
Each of these approaches produces a different idea of what temperature is.   So "temperature" as the term is used by scientists is one of several different things.    The only good news is that for most macroscopic objects and applications in most situations, they all behave similarly and it's even possible to identify that any one scale is measuring almost exactly the same sort of thing as another scale.   Note that we tend to say "scale" and it sounds like the only difference is in the choice of units for measurement but that's not what is actually happening.   The true situation is that slightly different things or different properties are actually being quantified by each scale.   For example, one scientist is measuring the height of a person,  another scientist is measuring the length of their shadow, outside of the standard regime or situation (so that's a macroscopic object with millions of particles and temperatures high enough so that QM is unimportant) one measurement is not just a fixed conversion factor  times by  the other measurement, they just are very different and non-comparable things.

     In the post 2019 definition of temperature in kelvin,  the temperature of an object is not required to be a measure of any physical thing you could actually observe.  Instead it is a much more abstract quantity,  it is the average k.e. that a particle would have in the model.   It doesn't matter whether you could actually observe particles and measure them to have kinetic energies like that.  Whatetver the situation, it is understood that temperature is describing something - an average kinetic energy - that might exist only in theory, in the model.
 
     However, this ("temperature" quantity) emerges as a real physical property or characteristic of a macroscopic object, statistical averages hold,  all of statistical mechanics and thermodynamics will hold well for that object etc. etc.

    Overall then, it's possible to say "temperature (as defined by the modern kelvin scale)" is an entirely theoretical quantity, however it becomes a real emergent property which can be assigned to macroscopic objects.  That is to say, at least to a high level of accuracy, it can be considered and modelled as a continuous parameter, T, that a macroscopic body has.

Best Wishes.

[Bored chemist]

The thing is that temperature also has to follow the zeroth law of thermodynamics.
And that means that heat capacities really do very with temperature.
The QM effects of rotation are usually safely ignored (maybe not for H2 at low temps- which is becoming increasingly important)- but the QM effects are also manifest in the variations of heat capacity with temperature.
The vibrational effects are not something you can ignore.
There's currently  a thread on this site asking about the maximum temperature you can reach with a charcoal fire.
If you ignore the (quantised) vibrational heat capacity in that calculation, you get a seriously wrong answer. 
I'd have to check, but I think you need to start looking at the electronic transitions too.