Scrutable

My basic understanding of temperature is that it represents the amount of energy that is "floating about" in a given place.

Why, then, are the units of temperature not joules per cubic metre?

Different materials having different specific heat capacity?

sTeamTraen wrote: Sat Nov 16, 2019 8:16 pm My basic understanding of temperature is that it represents the amount of energy that is "floating about" in a given place.

Why, then, are the units of temperature not joules per cubic metre?

Thermodynamics was never my strongest subject, but I believe it is the distribution of the energy amongst the particles.
Consider a bunch of particles all travelling to the left with the exact same high velocity. They would have a lot of energy but would be really cold.

From here

Heat is the total energy of molecular motion in a substance while temperature is a measure of the average energy of molecular motion in a substance. Heat energy depends on the speed of the particles, the number of particles (the size or mass), and the type of particles in an object.

sTeamTraen wrote: Sat Nov 16, 2019 8:16 pm My basic understanding of temperature is that it represents the amount of energy that is "floating about" in a given place.

Why, then, are the units of temperature not joules per cubic metre?

Most generally, temperature is the rate of change of enthalpy with a change in entropy.

In an ideal gas, it's also a measure of the kinetic energy of the particles of the gas, but that's a very specific case.

dyqik wrote: Sat Nov 16, 2019 10:30 pm

sTeamTraen wrote: Sat Nov 16, 2019 8:16 pm My basic understanding of temperature is that it represents the amount of energy that is "floating about" in a given place.

Why, then, are the units of temperature not joules per cubic metre?

Most generally, temperature is the rate of change of enthalpy with a change in entropy.

In an ideal gas, it's also a measure of the kinetic energy of the particles of the gas, but that's a very specific case.

It's specific to that case because an ideal monoatomic gas can only express energy in terms of the kinetic energy of the atoms of the gas (assuming we are not yet at the very high temperatures required to excite a significant proportion of the electrons in the atoms to higher levels). To have a temperature, the energy profile of the atoms needs to follow a certain distribution.

Each of the three dimensions in which an atom can move is a "degree of freedom". If we have a diatomic gas, then vibrations and rotations of the molecule add other degrees of freedom (usually at room temperature we need to consider two axes of rotation, but we don't yet consider the vibrations, because they require an even higher temperature to become activated). If we have a solid, then the vibrations of the atoms in the structure of the material make up degrees of freedom. Whatever you have, if it has something which it does more of when there is more energy in it, that's a degree of freedom, in this context at least.

That something is at a temperature T when each degree of freedom contains (at a suitably atomic/molecular level) ½kT of energy (or if it's a whole mole of a gas, ½RT), distributed statistically in the appropriate manner. So there's your link with energy, but if the distribution isn't a "thermal" one (or the system is so simple that it can't be described statisically) then the system just doesn't have a well-defined temperature.

^That is about what I remember from my degree. So how does that apply to places where there are only quanta of em radiation flying about? I often read about the temperature of deep space and wonder what it is. Is it something to do with black body radiation?

greyspoke wrote: Sun Nov 17, 2019 10:47 am ^That is about what I remember from my degree. So how does that apply to places where there are only quanta of em radiation flying about? I often read about the temperature of deep space and wonder what it is. Is it something to do with black body radiation?

If you consider a cubic box with shiny sides, you can consider photons bouncing around inside it. If they too have a thermal spectrum, because photon energy is proportional to their frequency, then when you put a small hole in the side of the box and consider the flux of photons coming out, you will get a black body radiation spectrum.

greyspoke wrote: Sun Nov 17, 2019 10:47 am ^That is about what I remember from my degree. So how does that apply to places where there are only quanta of em radiation flying about? I often read about the temperature of deep space and wonder what it is. Is it something to do with black body radiation?

That's what I recall (probably quite poorly) from my thermodynamics - the temperature of a vacuum is a meaningless idea, there being no mass to have a temperature.

shpalman wrote: Sun Nov 17, 2019 10:52 am

greyspoke wrote: Sun Nov 17, 2019 10:47 am ^That is about what I remember from my degree. So how does that apply to places where there are only quanta of em radiation flying about? I often read about the temperature of deep space and wonder what it is. Is it something to do with black body radiation?

If you consider a cubic box with shiny sides, you can consider photons bouncing around inside it. If they too have a thermal spectrum, because photon energy is proportional to their frequency, then when you put a small hole in the side of the box and consider the flux of photons coming out, you will get a black body radiation spectrum.

So the radiation in space is a combination of the cosmic microwave background and stuff coming from stars. Will any of that have the same distribution as black body radiation? And if it doesn't have a ?Boltzmann distribution, does temperature still mean something/the same thing?

Yes, the CMB radiation has a Boltzmann distribution with a peak at 3K.

philip wrote: Sun Nov 17, 2019 11:25 am Yes, the CMB radiation has a Boltzmann distribution with a peak at 3K.

2.72548±0.00057 K

It's probably the most perfect blackbody spectrum available, and so also thing that most follows the idealized definition of temperature. I think there's an XKCD for that

The 0.57 mK includes variation in CMB temperature in different places due to physics in the early universe, not just measurement errors.

Thanks peeps. Does it have to be randomly oriented radiation like the cmb, rather than a beam of light going somewhere? (By analogy with particles I suppose.)

greyspoke wrote: Sun Nov 17, 2019 3:32 pm Thanks peeps. Does it have to be randomly oriented radiation like the cmb, rather than a beam of light going somewhere? (By analogy with particles I suppose.)

As long as there's a defined set of photons with an energy distribution, you can define a temperature for it. The distribution doesn't have to be Maxwellian, as you can use T = dq/dS to define temperature.

This is also how you get to negative temperatures in inverted populations - e.g. in laser systems. If you have an inverted two level system with higher occupation of the upper state than the lower state, increasing the entropy requires that you take energy out of the system, so dq/dS is negative.