[hamdani yusuf (OP)]

It's possible to obtain a sample of a gas where most of the atoms are in an excited state.

How would you do that without increasing the temperature?

[hamdani yusuf (OP)]

You first...

Here.

If 2 Joules of energy is absorbed by a mass-spring system, on average 1 Joule will be in the form of kinetic energy, while 1 Joule is in potential energy.
For comparison, if the mass is put into a perfectly elastic box without attached to spring, 2 Joules of energy will be in the form of kinetic energy and 0 potential energy.
As analogy for temperature, the mass-spring system has twice heat capacity as the spring-less system. We can add the same amount of energy to both systems, but only half is manifested as kinetic energy in the first system, which is comparable to lower increase of temperature.

The widest range of temperature measurement methods I know is by ideal gas law, which can be approached practically using monoatomic noble gases, although they start to deviate at high enough temperature where the gas starts to ionize. The electromagnetic interactions makes the energy no longer strictly in kinetic form.

[Bored chemist]

It's possible to obtain a sample of a gas where most of the atoms are in an excited state.

How would you do that without increasing the temperature?

The scientific answer is "Who cares? Trust me; it's possible".
But here's the basics.

"The next step is "state selection"—in order to get some stimulated emission, it is necessary to create a population inversion of the atoms. This is done in a way that is very similar to the Stern–Gerlach experiment. After passing through an aperture and a magnetic field, many of the atoms in the beam are left in the upper energy level of the lasing transition. From this state, the atoms can decay to the lower state and emit some microwave radiation.

from
https://en.wikipedia.org/wiki/Maser#Some_common_types

[Bored chemist]

For comparison, if the mass is put into a perfectly elastic box without attached to spring, 2 Joules of energy will be in the form of kinetic energy and 0 potential energy.

The fact that you specify a perfectly elastic box means that it spends some of its time stretched.
And at that point you have stored potential energy in it.

[Eternal Student]

Hi.

Let's make the ball and the floor make perfectly elastic collision. On average, the ball has 5 Joule of kinetic energy and 5 Joule of potential energy.

  and also this section....

If 2 Joules of energy is absorbed by a mass-spring system, on average 1 Joule will be in the form of kinetic energy, while 1 Joule is in potential energy.

    is slightly inaccurate.

You need to be clearer about what you are averaging over.  Usually it would be an average over time - but that won't work in these examples.     The bouncing ball has the lowest speed at the top of its bounce and so spends much more time there,   with  high potential energy and low kinetic.   Similarly the  mass on a spring has the lowest speed when the spring is most extended.        Considering an average over time, the total energy in these systems is not equally partitioned between kinetic and potential,  much more than half of it will be potential energy.

    The equipartition theory is actually quite difficult to derive from fundamental principles.   I do not know or concern myself too much with those details.   For real life systems we have a reasonable assumption of ergodicity or randomness and this is enough to allow the equi-partition theorem, with just simple time-averages, to hold.
    For the systems you have suggested a different average, an average over phase space (the space of generalised momentum and position) should still satisfy the equi-partition result.  It doesn't reduce to an average over time because the property of ergodicity is lacking.   Ergodicity is explained in Wikipedia here:   https://en.wikipedia.org/wiki/Ergodicity.    Along with a general discussion of the equipartition theorem here:  https://en.wikipedia.org/wiki/Equipartition_theorem#General_formulation_of_the_equipartition_theorem
 and finally a guide to its derivation here:    https://en.wikipedia.org/wiki/Equipartition_theorem#Derivations

Best Wishes.

[hamdani yusuf (OP)]

The scientific answer is "Who cares? Trust me; it's possible".

"Trust me" is never a scientific answer.

[hamdani yusuf (OP)]

The fact that you specify a perfectly elastic box means that it spends some of its time stretched.

It merely means that kinetic energy is preserved.

[hamdani yusuf (OP)]

You need to be clearer about what you are averaging over.  Usually it would be an average over time - but that won't work in these examples.     The bouncing ball has the lowest speed at the top of its bounce and so spends much more time there,   with  high potential energy and low kinetic.   Similarly the  mass on a spring has the lowest speed when the spring is most extended.       

You need to pay attention to the accompanying pictures.

For comparison, if the mass is put into a perfectly elastic box without attached to spring, 2 Joules of energy will be in the form of kinetic energy and 0 potential energy.

This means remove the spring and add an elastic wall to the right of the mass.

[hamdani yusuf (OP)]

The bouncing ball has the lowest speed at the top of its bounce and so spends much more time there

Time ratio between top and bottom depends on the height difference, maximum speed, and gravitational acceleration. In ordinary lab desktop experiments with gas at near STP, it's close to 1.

[Bored chemist]

It merely means that kinetic energy is preserved.

How?

[hamdani yusuf (OP)]

The bouncing ball has the lowest speed at the top of its bounce and so spends much more time there,   with  high potential energy and low kinetic.

In this model, the top of the container would have lower temperature than the bottom. Let's say that two identical piezoelectric generators are placed on top and bottom of the container. Bottom generator will produce more power than the top generator.
This can be demonstrated by filling a tall container with heavy gases. The container should be thermally isolated from the outside. The heavier the gas, the shorter the container can be while still showing temperature difference.

[hamdani yusuf (OP)]

It merely means that kinetic energy is preserved.

How?

That's the definition of elastic collision. It doesn't depend on how long/short the collision happens, as long as the kinetic energy of the system is preserved.

[Bored chemist]

It merely means that kinetic energy is preserved.

How?

That's the definition of elastic collision. It doesn't depend on how long/short the collision happens, as long as the kinetic energy of the system is preserved.

The thing about "elastic" is that, by definition, it stretches.
And the only reason that kinetic energy is conserved is that it ie loaned to the surface as potential energy and then returned as kinetic energy on the rebound.
So, once you know how the kinetic energy is conserved, you realise that, ironically, it isn't. Briefly, it is converted to potential energy.

Did you not realise that was why I asked "How?" ?

Instead of answering the question, you posted some nonsense about duration.

[hamdani yusuf (OP)]

The thing about "elastic" is that, by definition, it stretches.
And the only reason that kinetic energy is conserved is that it ie loaned to the surface as potential energy and then returned as kinetic energy on the rebound.
So, once you know how the kinetic energy is conserved, you realise that, ironically, it isn't. Briefly, it is converted to potential energy.

Did you not realise that was why I asked "How?" ?

Instead of answering the question, you posted some nonsense about duration.

The time spent by a particle to have potential energy can be much smaller than the time it spent to have kinetic energy. That's why energy of ideal gas is almost exclusively in the form of kinetic energy, which makes its specific heat capacity minimum.

[alancalverd]

In an ideal gas, all collisions are perfectly elastic, by definition, and therefore the rebound is instantaneous..

[Bored chemist]

In an ideal gas, all collisions are perfectly elastic, by definition, and therefore the rebound is instantaneous..

Only for a monatomic gas.
Molecular gases vibrate.
And, in any event, you are in danger of solving the problem for a spherical horse in a vacuum.

[alancalverd]

Which is why we distinguish between real gases and ideal gases. In the GCSE fantasy world of weightless strings and frictionless pulleys, all gases are composed of tiny billiard balls and PV = RT for ever.

[hamdani yusuf (OP)]

Which is why we distinguish between real gases and ideal gases. In the GCSE fantasy world of weightless strings and frictionless pulleys, all gases are composed of tiny billiard balls and PV = RT for ever.

We solve problems starting from the most basic and simple version. When the result is satisfactory, we must expand it to more general and realistic situation by introducing other factors.
In real life, we apply temperature measurement not only to ideal gas. We also have to measure temperature of molecular gases, liquids, solids, and plasmas. We must also measure fluid temperature in various pressure. Plotting the heat capacity in various temperature and pressure can give us insight on the microscopic states shown as temperature of objects.

[alancalverd]

And when we look at heat transfer between objects, we recognise that there is none between bodies at the same temperature, from which we infer that temperature is a measure of the mean internal kinetic energy of a body.

[Bored chemist]

from which we infer that temperature is a measure of the mean internal kinetic energy of a body