Can temperature be expressed in terms of mass, displacement, time, and charge?

[Izzhov]

It is possible to put almost every unit in physics in terms of a product of integer powers of mass (kg), displacement/distance (m), time (s), and charge (q). Here are some examples:
$$\begin{array}{c} speed = m^1 \times s^{-1} \\ acceleration = m^1 \times s^{-2} \\ jerk = m^1 \times s^{-3} \ \\ momentum = kg^1 \times m^1 \times s^{-1} \\ force = kg^1 \times m^1 \times s^{-2} \\ energy = kg^1 \times m^2 \times s^{-2} \\ power = kg^1 \times m^2 \times s^{-3} \\ pressure = kg^1 \times m^{-1} \times s^{-2} \\ current = q^1 \times s^{-1} \\ voltage = kg^1 \times m^2 \times q^{-1} \times s^{-2} \\ resistance = kg^1 \times m^2 \times q^{-2} \times s^{-1} \end{array}$$
...and so on. My question is: Is there any way to put units of temperature (degrees) into this format?

 

[rbj]

temperature is essentially the amount of unordered kinetic energy per particle per degree of freedom. it's a measure of energy and the conversion factor is the Boltzmann constant. Temperature can be removed as a physical quantity, expressing it in terms of the other base physical quantities (time, mass, length) but so can electric charge (as they do with the cgs system of units). however, whereas i don't consider temperature to be a fundamentally different physical quantity (it's just a measure of energy), i do consider electric charge to be a fundamentally unique or separate dimension of physical "stuff".i might recommend Wikipedia at Fundamental unit, Dimensional analysis, Planck units

 

[Izzhov]

temperature is essentially the amount of unordered kinetic energy per particle per degree of freedom. it's a measure of energy and the conversion factor is the Boltzmann constant. Temperature can be removed as a physical quantity, expressing it in terms of the other base physical quantities (time, mass, length) but so can electric charge (as they do with the cgs system of units). however, whereas i don't consider temperature to be a fundamentally different physical quantity (it's just a measure of energy), i do consider electric charge to be a fundamentally unique or separate dimension of physical "stuff".i might recommend Wikipedia at Fundamental unit, Dimensional analysis, Planck units

How would one express temperature in terms of the other units, then?

I looked it up on Wikipedia: It defined units of temperature in terms of energy and entropy. Then I looked up entropy: it defined the units in terms of temperature. Neither entry mentioned the fundamental units. (The fundamental units article didn't help, either, because they treat temperature as a fundamental unit in itself!)

 

[StatMechGuy]

I personally measure entropy as a unitless quantity, and temperature in the units of energy. However, units don't tell you what something means, necessarily.

Mehran Kardar's notes for Stat. Mech. I on the MIT CourseWare website have a great discussion of temperature from a microscopic viewpoint. Basically, he shows that temperature can be thought of as a constraint between all the parameters between two systems in thermal equilibrium. In my mind, this is a fruitful way of looking at temperature, since it's a very physically grounded defintion.

 

[Izzhov]

Basically, he shows that temperature can be thought of as a constraint between all the parameters between two systems in thermal equilibrium.

Is there any chance you could put that in somewhat simpler terms...? I know what thermal equilibrium means, and I know what a system is, but I don't quite understand the "constraint between all the parameters" part.

 

[Izzhov]

however, whereas i don't consider temperature to be a fundamentally different physical quantity (it's just a measure of energy), i do consider electric charge to be a fundamentally unique or separate dimension of physical "stuff".

This is why I use charge as one of my fundamental units instead of the traditional SI unit, amperes.

 

[ShawnD]

How would one express temperature in terms of the other units, then?

I looked it up on Wikipedia: It defined units of temperature in terms of energy and entropy. Then I looked up entropy: it defined the units in terms of temperature. Neither entry mentioned the fundamental units. (The fundamental units article didn't help, either, because they treat temperature as a fundamental unit in itself!)

Temperature sort of is a fundamental unit. The correlation between temperature and energy is sketchy because things can sponge a different amount of energy at different temperatures. That chart says you might be able to change the temperature by 1 degree using 1J of energy at a given temperature, then you might need 1.2J to get the same change at a higher temperature, then 1.5, then 1.6 etc etc.

You can't directly correlate temperature to an exact amount of energy without knowing what the material is, and that's why I think temperature is a fundamental unit.

 

[Izzhov]

Temperature sort of is a fundamental unit. The correlation between temperature and energy is sketchy because things can sponge a different amount of energy at different temperatures. That chart says you might be able to change the temperature by 1 degree using 1J of energy at a given temperature, then you might need 1.2J to get the same change at a higher temperature, then 1.5, then 1.6 etc etc.

You can't directly correlate temperature to an exact amount of energy without knowing what the material is, and that's why I think temperature is a fundamental unit.

So basically what you're saying is, the correlation between temperature and energy in a given substance is an intrinsic property of the substance, and therefore it should be treated as a fundamental unit?

 

[ShawnD]

Yes that is exactly what I am saying. You can put it in terms of entropy and enthalpy but in the end you still need to know what the material is.

 

[Mech. Engineer]

can you express length unit (m) in terms of the other units?



the same with temp.

 

[Izzhov]

can you express length unit (m) in terms of the other units?



the same with temp.

Yes, you can. Distance equals velocity times time. Also, temperature equals Heat divided by (mass times specific heat). However, I still consider distance and temperature to be fundamental units.

 

[Hootenanny]

Yes, you can. Distance equals velocity times time.

Yes, but what are the units of velocity? That's right meters per second; you can't use a unit to define itself!

Also, temperature equals Heat divided by (mass times specific heat).

Erm, no it doesn't. That would be change in temperature, not actually temperature.

 

[Izzhov]

Yes, but what are the units of velocity? That's right meters per second; you can't use a unit to define itself!

Erm, no it doesn't. That would be change in temperature, not actually temperature.

That's exactly why I still consider each of those units to be fundamental.

 

[StatMechGuy]

Is there any chance you could put that in somewhat simpler terms...? I know what thermal equilibrium means, and I know what a system is, but I don't quite understand the "constraint between all the parameters" part.

Basically, the argument is as follows.

System A can have its microscopic configuration specified by a set of parameters A_1, A_2, etc. The same can be done for system B and system C. If A and B are in thermal equilibrium, that this indicates that there is a "constraint equation" (same concept as with Lagrangian mechanics) f(A_1, A_2, ... B_1, B_2, ...) = 0.

He uses this argument to prove the zeroth law of thermodynamics from microscopic considerations, and later proves that this constraint serves the function of temperature, and therefore calls it as such. I can't really think of a "simpler" way of putting this, it's a little abstract to begin with.

Here is the url for those particular lecture notes: http://ocw.mit.edu/NR/rdonlyres/Physics/8-333Fall-2005/BC713F26-6A2D-499A-B6DB-657804F9C32A/0/lec1.pdf"

 

[StatMechGuy]

Yes, but what are the units of velocity? That's right meters per second; you can't use a unit to define itself!

Erm, no it doesn't. That would be change in temperature, not actually temperature.

In the modern system of measuring units, time is determined in terms of the frequency of oscillation for a particular element of cesium, and a meter is then defined as the distance light travels in some absurdly short period of time. So technically, the meter is defined in terms of a velocity, but it's a universally constant velocity multiplied by a constant time (in the same inertial frame).

 

[pkleinod]

I believe the answer to the original post of Izzhov is that
temperature is unitless; i.e. the degree is really not a fundamental unit but
just determines the numbers that are used in a particular temperature scale.
Temperature is a measure of whether there will be a net flow of heat from one
body to another with which it is in thermal contact. Any
scale of temperature must satisfy the zeroth law of thermodynamics,
which effectively says that there will be a net heat flow from a hot
body to a cooler body. The choice of scale is not determined from this
law.

The temperature of a system can be obtained by measuring the product (PV) of a
fixed amount of any gas in contact with the system at a series of
pressures and then extrapolating the product to zero pressure. P is the
pressure and V the volume of the gas. This is
the ideal-gas scale of temperature (See Giauque:
Nature, 143, 623, (1939)) and was
adopted by the International Unions of Physics and Chemistry:

T=273.1600 lim_(P-->0)(PV)_T/lim_(P-->0)(PV)_t.p.

Here the subscript t.p. refers to the triple point of water,
The number 273.1600 is chosen to make the degree of temperature the same size
as that of the centigrad (Celsius) scale. Temperature is related to energy
via the equation for the ideal gas: PV = nRT, where n is the number of moles
of gas and R is the gas constant, which is a fundamental constant:
R = 8.314 J K^-1 mol^-1.

The thermodynamic temperature enters into the definition of entropy
as an integrating factor. It can be shown from standard thermodynamics
arguments that in fact the thermodynamic temperature may be chosen to be
identical to the ideal-gas scale of temperature. This scale does not
depend on the properties of anyone substance used in measuring the
temperature. In particular, the specific heat of a substance plays no
role in the definition of the temperature scale.

 

[peaceharris]

Temperature is a measure of whether there will be a net flow of heat from one
body to another with which it is in thermal contact. Any
scale of temperature must satisfy the zeroth law of thermodynamics,
which effectively says that there will be a net heat flow from a hot
body to a cooler body. The choice of scale is not determined from this
law.

Do you agree with people who say that the thermosphere is very hot?
(I.E this http://members.tripod.com/atmosphere_guys/thermosphere.html" says, "The actual temperature in the Thermosphere can reach as high as 2000º C"

But if you take a jar of boiling water at 100C to the thermosphere, I am sure that heat will flow from the jar to the thermosphere and it will become ice.

I believe that the thermosphere (and outer space in general) has high speed particles because high speed particles can escape the gravitational field of planets and stars. The Maxwell distribution of velocities in a gas predicts that there will always be a small percentage of particles having a velocity more than the escape velocity.

But these particles rarely collide with each other, and it is meaningless to say that their temperature is high because of their high kinetic energy. Would you agree?

 

[pkleinod]

Do you agree with people who say that the thermosphere is very hot?

Dear Peaceharris,
Your have posed four questions:
1) Do I agree that the thermosphere is hot?
2) Would a jar of boiling water placed in the thermosphere freeze to ice?
3) Do I agree that particles in outer space have high velocity because
they are the ones on the tail of the Boltzmann distriubution that
lies aove the escape velocity?
4) Is it meaningless to ascribe a high temperature to particles having a
high velocity?

Oh dear, this post may be a bit long! As a preface to my answers, let me
recall that my reply to the original post about the units of temperature
was couched in the terminology of classical thermodynamics, in which the state
functions (e.g. energy, entropy, enthalpy, fugacity, free energy, etc.)
refer to systems in equilibrium. Moreover, I avoided all mention of
statistical mechanics (which is just a mechanistic interpretation of
classical thermodynamics) because it was not needed. To answer your questions,
one needs to consider the distribution functions that arise in statistical
mechanics, where temperature can be regarded as a parameter in a function
determining the distribution of some quantity (e.g velocity or energy or
quantum state etc) among a number of particles. An important remark here is
that if temperature is so regared, then it is nonsense to ascribe a
temperature to any particular particle; i.e. we must always bear this in
mind. So now to my answers.

1) The thermosphere is not in thermodynamic equilibrium (and never
will be in my lifetime!), so to speak of a temperature in the usual
sense is meaningless. So what do people mean by hot (2000 C in this
case)? According to the text in the web-site, "temperature is a measure
of how fast particles move" (This is of course not a very good definition.
It is like saying that money is a measure of the value of cheese.), so
they must be using the velocity distribution of the particles to define
a temperature, i.e. they are asking "What temperature would a Boltzmann
distribution have to have in order to fit the observed
distribution of velocities?" The answer is apparently T=2000 C. I
can agree with this kind of parlance but one must realize that this
is not the temperature. If a system is not in equilibrium, there can
be as many temperatures as
there are quantities for which a distribution can be determined, and
all of these can be different! Even negative temperatures can
be observed by playing this game! (I can give you an example in
another post if you are interested.) At equilibrium, all of these
temperatures would be equal.

2) Are you so sure that boiling water would freeze in the thermosphere?
I have no idea, not being an atmospheric physicist but consider this:
You are on your way to a wedding at 10:00 am dressed in your best
black suit. An alien specimen collector snaps you up, puts you in
a big closed jar of air and places you in the thermosphere directly
above the spot upon which you were walking. Will you freeze? I think
not---you will probably be fried in the sunshine. And if instead, you
were plucked from the street on returning from the wedding at 11 pm? Then
I think you would freeze in the dark shadow of the earth. Maybe this
latter situation was your point: the "hot" particles would not prevent
your freezing. True, but not because they are not hot but because there
simply aren't enough of them---they would have to compete with the
rate at which the boiling water is losing energy owing to radiative
cooling (infra-red radiation).

3) Suppose you are in the thermosphere and you measure the velocity
of those molecules leaving the Earth that reach you. Yes, you would
find some molecules with high velocity but you would find many many
more with low velocity, namely those that do not have
escape velocity but which are fast enough to have reached your
position (remember that gravity is a very long range force).
(In order to have reached you, they had to overcome the gravitational
force and have therefore a smaller velocity than when they started
their journey)
If you play the fitting game with your observed distribution, you
might find a "temperature" which was not too different from the
original temperature of the molecules nearer the earth. Note that
many of the molecules reaching you would have zero velocity.

But there is an even more important consideration: are there any
other processes that could produce fast particles? I can think of
one at least. The web-site mentioned that the oxygen and nitrogen
molecules were good at converting "heat" into velocity. How
do they do this? One way would be for a molecule to absorb radiation
from the sun and thus go from its ground state into a highly
excited electronic state. If the latter were not stable with respect
to molecular dissociation, the two atoms (e.g two N atoms from
nitrogen) would then fly apart with a high velocity. I don't know if
such a process is important in the thermosphere but it is certainly
well-known in the laboratory and would merit your investigation before
you place too much faith in your escape velocity theory.

4) I agree with you. It is meaningless to talk of the temperature of a
particle (in the context of our discussion here).

 

[peaceharris]

2) Are you so sure that boiling water would freeze in the thermosphere?

I think we can estimate the final steady state temperature of a sealed jar of water in the thermosphere by making certain assumptions:

Assume that your jar is spherical with radius r, and it absorbs all energy falling on it, and it is a perfect thermal conductor, and the walls of this jar radiates like a blackbody. The heat absorbed by the jar depends on the heat it gets from the sun and the earth.

Case 1: Its 12 noon.

Total power of the sun= 4e26 Watts
Power per unit area at a distance D from the sun= 4e26/(4*pi*D^2)
Power incident on the jar from the sun=4e26 * pi * r^2/ (4*pi*D^2)

The power radiated out from the walls of this jar using Stefan-Boltzmann's law =s * 4 *pi*r^2 * T^4 (where s is Stefan's constant)

Total power radiated by Earth assuming the Earth radiates like a black body, with a uniform surface temperature of 300K = Stefan's constant * 4 *pi*R^2 * 300^4 (where R is the radius of the earth)

Total power radiated by Earth per unit area at the thermosphere 100km from the Earth's surface (at the thermosphere)=Stefan's constant * 4 *pi*6360000^2 * 300^4 /(4*pi*6460000^2) (I have assumed R=6360000m)

Power incident on the jar from the earth=Stefan's constant * 4 *pi*6360000^2 * 300^4 *pi * r^2 /(4*pi*6460000^2)

At steady state, power incident on the jar from sun and Earth = power radiated out, so we can solve for T.

At 12 noon, based on this arithmetic, T is about 301K

Case 2: Its 12 midnight, there's no power from the sun:
At steady state, power incident from Earth = power radiated out

At 12 midnight, based on this arithmetic, T is about 210K

I don't think the energy imparted to the jar from collision of high speed particles to the jar is going to make much difference to this number. In any case, the final temperature of the jar initially with boiling water at 100C will reduce, implying that heat has flowed out. So I think people should stop saying that the thermosphere is hot or is at 2000K, because there will always be a net flow of heat from the jar of boiling water of 100C to the thermosphere.

 

[pkleinod]

Peaceharris: Yes, I fully agree that the effect of the high speed particles on your temperature estimate would be negligible.