[Bored chemist]

What's your idea of the cut off for calling something "umpteen"?
Is it 1,2,3 or 4?

[hamdani yusuf (OP)]

Here is a more complete table of molar heat capacity I compiled from NIST website.

Temp (K)      Hydrogen   Deuterium   Helium   Argon   Radon
300               28.85        29.19           20.79    20.79    20.79
1000             30.20        31.64           20.79    20.79    20.79
3000             37.09        38.16           20.79    20.79    20.79
6000             41.97        42.25           20.79    20.79    20.79

From the table we can conclude that increase of temperature also increases the portion of rotational and vibrational movements in kinetic energy of diatomic gases. In noble gases, those types of motion are virtually non-existent.

[hamdani yusuf (OP)]

Here is a more complete table of molar heat capacity I compiled from NIST website.

Temp (K)      Hydrogen   Deuterium   Helium   Argon   Radon
300               28.85        29.19           20.79    20.79    20.79
1000             30.20        31.64           20.79    20.79    20.79
3000             37.09        38.16           20.79    20.79    20.79
6000             41.97        42.25           20.79    20.79    20.79

From the table we can conclude that increase of temperature also increases the portion of rotational and vibrational movements in kinetic energy of diatomic gases. In noble gases, those types of motion are virtually non-existent.

If we compare the results for hydrogen and deuterium, we get that particle's mass affect the portion of rotational and vibrational movements in kinetic energy of diatomic gases. But the mass doesn't affect molar heat capacity of noble gases whose motions are restricted to almost exclusively translational.

From those results, we can infer that temperature is proportional to particle's mass and square of particle's speed. From previous information we also obtain that different type of motions contribute differently to the temperature of a system. Thus,
T=C.∑m.vn².εn
where
C is a proportionality constant.
m is particle's mass
vn is particle's speed in corresponding degree of freedom.
εn is effectiveness of each degree of freedom to affect system's temperature.

In case of noble gases, we can put vn close to 0 for rotational and vibrational motion.
In diatomic gases, εn for rotational and vibrational motion don't seem to be constant, but they're affected by temperature instead.

[Bored chemist]

In diatomic gases, εn for rotational and vibrational motion don't seem to be constant, but they're affected by temperature instead.

It's one aspect of the quantisation of energy.

[hamdani yusuf (OP)]

In diatomic gases, εn for rotational and vibrational motion don't seem to be constant, but they're affected by temperature instead.

It's one aspect of the quantisation of energy.

How can energy quantisation help to explain the phenomenon I described above?

[Bored chemist]

The typical energy of a molecule depends on temperature- it's 1/2 kT per degree of freedom.
In a classical world,you could have the molecules rotating at any speed, including very low ones.
And so, even at very low temperatures the heat capacity would include a contribution from rotation.
But in the quantum world, only certain rates of vibration are allowed.
So, unless kT is big enough, it will not be able to excite the rotation of the molecule and that will reduce the heat capacity of cold molecules.
The same is also true of vibrations and of electronic excitations.

[hamdani yusuf (OP)]

Temp (K)        Helium       Hydrogen      Deuterium      Nitrogen   Oxygen   Fluorine
300                  20.79          28.85          29.19          29.12             29.39     31.38
1000                20.79          30.20          31.64          32.69             34.86     37.09
3000                20.79          37.09          38.16          37.03             39.87     37.54
6000                20.79          41.97          42.25          38.27             44.39     28.99

Let's compare heat capacities of other light diatomic gases: N2,O2 & F2.
Nitrogen with triple bond has low heat capacity at low temperature, but increase more steadily.
Oxygen with double bond increase more heat capacity as temperature raised.
Fluorine with single bond initially increase heat capacity quicker, but then decline at high temperature.

 

[Bored chemist]

Oxygen has a relatively low lying excited electronic state
https://en.wikipedia.org/wiki/Singlet_oxygen
which complicates things.
I'm not sure how hot you have to get fluorine before it dissociates into two atoms.

[hamdani yusuf (OP)]

I guess that the decline of heat capacity of F2 above 2000K indicates that some of them has dissociated to become monoatomic, thus losing some degrees of freedom. At higher temperature, more of them are dissociated, so their heat capacity gets closer to noble gases.

[hamdani yusuf (OP)]

In vibrational motion, the kinetic energy is continuously exchanged with potential energy. So for a collection of particles with random phases of vibration, only some part of the system's total energy is manifested as kinetic energy at any given time. That's why gases capable of vibrational motion shows higher heat capacity than those with pure translational motion.

[hamdani yusuf (OP)]

In vibrational motion, the kinetic energy is continuously exchanged with potential energy. So for a collection of particles with random phases of vibration, only some part of the system's total energy is manifested as kinetic energy at any given time. That's why gases capable of vibrational motion shows higher heat capacity than those with pure translational motion.

Similar thing may apply to rotational motion too. But it's not obvious how potential energy is related to rotational motion. Apparently, potential energy must be derived from force and distance.
Let's try to analyze this scenario.

Two identical spheres, rotating about the center of the string joining them. Because of the rotation, the string is under tension.

Each sphere has mass (m) of 1 kg.
The radius of the trajectory (r) is 1 meter.
Moment of inertia (I) is ∑m*r² = 2 kg.m².
The angular velocity is (ω) 1 rad/s,
Hence the tangential velocity (v) is 1 m/s.
String tension = centripetal force = m*v²/r = 1*1²/1 = 1 Newton.

Angular momentum (L) is I*ω = 2*1 = 2 kg.m²/s.
Kinetic energy (Ek) = ½*I*ω² = ½*2*1² = 1 Joule.

Suppose the central point of the string is made of a retractable mechanism with negligible mass. The mass of the string is also negligible.


Without significant external force, let's say using a timer inside, the lock mechanism of the retractable device is released.The string is then stretched so the radius of the new trajectory becomes 2 m.
The masses are conserved, 1 kg each.
Moment of inertia (I) is ∑m*r² = 1*2² + 1*2² = 8 kg.m².
Angular momentum (L) is conserved, still = 2 kg.m²/s.

The angular velocity is (ω) is L/I = 2/8 = 0.25 rad/s,
Hence the tangential velocity (v) is ω.r = 0.25*2 = 0.5 m/s.
Kinetic energy (Ek) = ½*I*ω² = ½*8*0.25² = 4/16 = 0.25 Joule.
String tension = centripetal force = m*v²/r = 1*0.5²/2 = 0.125 Newton.

We see there is a reduction of kinetic energy in the isolated system. If we assume that total energy is conserved, then there must be an increase in potential energy.

[hamdani yusuf (OP)]

We see there is a reduction of kinetic energy in the isolated system. If we assume that total energy is conserved, then there must be an increase in potential energy.

But if the sequence is reversed, we don't see identical result. If we start from 2 meter radius and finish with 1 m, we would need energy to retract the string, because some force is required to oppose the string tension. Where did the energy go in the previous case?

[hamdani yusuf (OP)]

Without significant external force, let's say using a timer inside, the lock mechanism of the retractable device is released.The string is then stretched so the radius of the new trajectory becomes 2 m.

Without energy dissipation, what we'll get is a reflection. The spheres will bounce back to their original radius.
When the lock is active, the spheres are moving in circular trajectory. But when it's released, they move linearly, tangent to the initial circle radius.
The distance of each spheres from the center vary between 1 and 2 meters. So the trajectory of each sphere, viewed from a stationary observer, is an isoscale triangle. The combined pattern will look like star of David.

[hamdani yusuf (OP)]

In reality, there is no string attached to each atoms in gas molecules. The centripetal force is provided by electrostatic and electrodynamic forces. Their strength depend on distance and velocity of the particles. These forces can store potential energy to the system.

[hamdani yusuf (OP)]

So far, we've discussing about temperature of gases.
Now I'd like to move on to other phases of matter. Particularly, I'm curious about what's the microscopic mechanism of emissivity factor ε. Black body radiation is just a special case where ε=1.
What's the minimum theoretical value of ε? What does it mean if a material has ε=0? What does it mean if a material has ε<0?
 

For surfaces which are not black bodies, one has to consider the (generally frequency dependent) emissivity factor ε(ν). This factor has to be multiplied with the radiation spectrum formula before integration. If it is taken as a constant, the resulting formula for the power output can be written in a way that contains ε  as a factor:

This type of theoretical model, with frequency-independent emissivity lower than that of a perfect black body, is often known as a grey body. For frequency-dependent emissivity, the solution for the integrated power depends on the functional form of the dependence, though in general there is no simple expression for it. Practically speaking, if the emissivity of the body is roughly constant around the peak emission wavelength, the gray body model tends to work fairly well since the weight of the curve around the peak emission tends to dominate the integral.

[Bored chemist]

Particularly, I'm curious about what's the microscopic mechanism of emissivity factor ε.

https://en.wikipedia.org/wiki/Kirchhoff%27s_law_of_thermal_radiation

[hamdani yusuf (OP)]

Particularly, I'm curious about what's the microscopic mechanism of emissivity factor ε.

https://en.wikipedia.org/wiki/Kirchhoff%27s_law_of_thermal_radiation

I've read the link, which starts with this description:

In heat transfer, Kirchhoff's law of thermal radiation refers to wavelength-specific radiative emission and absorption by a material body in thermodynamic equilibrium, including radiative exchange equilibrium.

A body at temperature T radiates electromagnetic energy. A perfect black body in thermodynamic equilibrium absorbs all light that strikes it, and radiates energy according to a unique law of radiative emissive power for temperature T, universal for all perfect black bodies. Kirchhoff's law states that:

For a body of any arbitrary material emitting and absorbing thermal electromagnetic radiation at every wavelength in thermodynamic equilibrium, the ratio of its emissive power to its dimensionless coefficient of absorption is equal to a universal function only of radiative wavelength and temperature. That universal function describes the perfect black-body emissive power.[1][2][3][4][5][6]
Here, the dimensionless coefficient of absorption (or the absorptivity) is the fraction of incident light (power) that is absorbed by the body when it is radiating and absorbing in thermodynamic equilibrium.

Unfortunately, I can't find the description of its microscopic mechanism, which will give us a way to increase or decrease the emissivity of a material at will.

[Bored chemist]

The mechanism of that law is the conservation of energy.
The microscopic explanation of colour is quite well known.


Choosing a material which is a different colour is easy.
Changing the colour of a material is sometimes difficult, but not impossible.
https://en.wikipedia.org/wiki/Interference_filter

If the world you worked in was cold enough, you could easily build metamaterials with variable properties (in the microwave region) and vary their absorbance and therefore the emissivity

[hamdani yusuf (OP)]

If I want to minimize heat loss from thermal radiation of a hot vessel, say 1000 °C, I must make the emissivity of its surface to near 0. It means that the surface radiates small amount of electromagnetic wave.
Theoretically speaking, 0 emissivity means that there's no electromagnetic radiation from the surface. But 1000 °C temperature of the vessel means that if a small metal plate at room temperature touches the vessel, it will receive thermal energy from the vessel through conduction, and its temperature will increase toward 1000 °C.

[Bored chemist]

If I want to minimize heat loss from thermal radiation of a hot vessel, say 1000 °C, I must make the emissivity of its surface to near 0.

If the emissivity is near zero, that means (by Kirchhoff's law) that the absorptivity is also zero.
 If the surface absorbs no light, then it must reflect it all.

That's why they silver the insides of thermos flasks.