The fundamental definition of temperature is

\(
T^{-1} = \frac{dS}{dE}
\)

How can I verify that the dimensions work in this equation?

 

Entropy is unit-less so that would give temperature units of energy, which seems right to me. The scales associated with temperature measurement are many and arbitrary.

 

IIRC, a change in temperature must be accompanied by a change in heat content. But temperature isn't a measure of heat, is it? (My thermodynamics texts are packed away...) Doesn't the energy unit drop out in the formula for specific heat?

 

Yeah, there seem to be two interpretations of energy that sometimes get confusing.

Heat is analogous to something like watt-seconds or work and temperature to volts or potential/state. I guess what I’m getting at are the microscopic vs. macroscopic.

The heat will change with the substance given a constant temperature. The heat of a given substance at a given temperature is a macroscopic view whilst the temperature is a microscopic view. The temperature of an ideal gas, for instance, would be proportional to the average translational energy of the gas molecules while the heat would be proportional to the total translational energy per quantity of the gas.

They are definitely related.

I see what you are saying though. Most of the kinetic or statistical physics derivations of temperature leave it unit-less. I think this is done as a matter of convenience though. It may have more to do with history and convention than anything else.

Maybe another case of the physics getting lost in the math.

 

I don't think your equation is correct.

Look up the second law of thermodynamics.

 

I'm pretty sure I've seen that before, where 1/T is equal to the rate of change of entropy with respect to energy. I don't think his equation is wrong, but I'm going by memory.

 

Look up the second law of thermodynamics.

Okay... so entropy is dimensionless, but the change in entropy is in Joules per degree? This is why I call it "thermodagnabbits."

 

It depends upon what energy Boks is talking about.

Gibbs energy ? Helmholtz energy? Total energy? Enthalpy?

The equation, as stated, only applies to reversible processes.

Temperature is an intensive state variable.

Using the second law as a definition of temperature iproduces a rather circular argument. One of the great successes of classical physics was showing the equivalence of temperature as defined by statistical thermodynamics(partition functions), mechanics (the gas laws)and classical thermodynamics (First and Second laws).

 

I'd have to go with total energy on this one. Enthaply is never denoted E (at least I've not seen it that way), but all that seperates totaly energy and Q is a constant.

I agree, it must be reversible.

 

Can I simply answer that the dimensions are correct because S is dimensionless, and T is a measure of energy?

 

But temperature is not a measurement of energy, is it?

It takes more energy to raise the temperature of a gram of water from 0C to 1C than a gram of copper at the same temperatures. And it takes more energy to raise the temperature of a metric ton of X from 0C to 1C than it does to raise the temperature of a milligram of X by the same amount.

 

But temperature is not a measurement of energy, is it?

Quite so.

Interesting to see Boks is prepared to put into the thread again, the answer is already given.

Temperature is a state variable, with its own dimensions - °K

Entropy is Joules per °K

Energy is Joules

So entropy divided by energy is joules per °K divided by joules, i.e (°K)\(^{-1}\)

 

Now that boks has his answer let me hijack the thread a little….

Quite so.

I don't want to get into an argument about this, but I disagree. Relating kinetic energy to pressure and using the ideal gas law you can easily show that the only thing separating temperature from an expression for average energy is an equal sign and an arbitrary constant. A constant that is free to be chosen at the whim of the person that etched their particular scale onto their thermometer.

Temperature is a state variable, with its own dimensions - °K

Yes, perhaps an aggregate state variable under normal conditions. State of what though, and under what conditions? Energy state? Of a balloon of gas, bowl of soup, salt crystal, single atom, single electron? All things that can give a temperature reading. So yes, I guess depending on you interpretation it is a state variable, but we’re getting into semantics.

Now as far as those dimensions are concerned, who should we ask for confirmation, Kelvin, Celsius, Fahrenheit, Rankine, Newton, Romer, Reaunmur, Delisle? Me, you? While some might agree on the spacing of the lines on the thermometer, they would agree on precious little else.

Entropy is Joules per °K

Agreed, these are the accepted units for S in the SI system and I apologise for my error on this, but if you take that arbitrary constant out, it is unit-less. I could just as easily mark my thermometer in joules per mole.

It boils down to this. There are many things in science that are open to interpretation.

 

But temperature is not a measurement of energy, is it?

It takes more energy to raise the temperature of a gram of water from 0C to 1C than a gram of copper at the same temperatures. And it takes more energy to raise the temperature of a metric ton of X from 0C to 1C than it does to raise the temperature of a milligram of X by the same amount.

It is, but not total energy, as I said beore, it's a measure of average enegry. Although, given some other specifics total energy can certainly be expressed as a function of temperature.

Energy can be and is expressed in many, many ways depnedng on how you are looking at a system.

Dogmatic minds don't evolve....

 

Relating kinetic energy to pressure and using the ideal gas law you can easily show that the only thing separating temperature from an expression for average energy is an equal sign and an arbitrary constant.

Feel free to develop this idea.

I only used MKS units because they are shorter to write than 'energy units' or 'temperature units' etc.
I agree that we need to choose a temperature scale, but surely this is no more or less arbitrary than the difference between the King Henry yard and the Napoleonic metre? Neither choice negates the fact that there is a fundamental property of nature we call 'length' and you can interchange scales by multiplication by a suitable number.

Constants such as Boltzman's constant in the gas law obviously depend arbitrarily on our choice of scale for their numerical value, but still posess dimensions or units in physics, to dimensionally balance the equations.

This is different from pure numbers such as angles, which have separate physical significance, despite being pure numbers,

Or numbers that are fixed and do not depend upon our choice of scale, eg Avogadro's number, which has units 1/(mole)
Edit no this is not a good example, sorry, the units are 1/(gram-moles) and so dependent upon the scaling of mass.
When you get beyond mechanics it becomes necessary to add further fundamental physical quantities to the basic MLT system. In electricity we normally add charge, and in thermodynamics, temperature.

Edit and even with mechanics there are limitations of the system for example,
stress is force divided by area. But there are different types of stress, shear and normal different animals with the same units.

We could, of course, choose different basic units, such as a unit of heat energy, and derive temperature from this, at the expense of extra complications in our mathematics.

 

I was going to leave this as an exercise for the readers, but..see the attached PDF. Sorry, but I don't know how to do super and sub scripts in this editor.

I agree with what you say, but since T is given arbitrary units, R is only there for that purpose. You could give T units of Joules/mole and you would not need R at all. R is one of those things in physics that hangs around for historical reasons. It's not really needed.

 

Thanks Bill, but what do I make of this?

It only relates to an ideal gas.

Ask the captain of the Titanic,
An iceberg contains a heck of a lot of energy, average energy per Kg or per cubic metre, but is pretty cold.

The best proof/explanation, I've seen, of why we need a physical property called temperature is in the opening chapter of the classic book

Chemical Equilibrium by Denbeigh.

However it's four pages long, too much to post here, but I could pm a scan.

The simplest I've seen is the A level statement that

"Temperature is that property which allows us to decide if two bodies are in thermodynamic equilibrium, in accordance with the zeroth law."

 

Forgive me, but I think you're stretching a bit studiot. We were talking temperature and how it relates to thermal energy, so I'm not going to include the relative kinetic energy of my particular ball of gas with respect to some other moving object, nor the latent energy in any chemical bonds, nor even the mass-energy equivalency relation'ship'.

All I'm trying to show is that temperature is a measure of energy. Specifically, the microscopic interpretation of average energy. Since temperature is, well temperature, my calculation using an ideal gas is valid. I chose an ideal gas because the calculations are trivial. I could have looked at the molecular motion in an oozing liquid, or the vibrating atoms in a crystal lattice or any other example of thermal energy and have come up with a similar result.

Edit - BTW, I have no argument against the need for a measurement of temperature.

 

you say, but since T is given arbitrary units, R is only there for that purpose. You could give T units of Joules/mole and you would not need R at all.

But don't different substances have different specific heats? Doesn't the same ratio of Joules/mole leave water at a lower Kelvin temperature than silicon?

 

But don't different substances have different specific heats? Doesn't the same ratio of Joules/mole leave water at a lower Kelvin temperature than silicon?

As studiot observed, Joules/mole are the units that correspond to the expression that relates to an ideal gas. It would be an appropriate approximation for the gaseous states of water and silicon. The expression would change considerably for a solid or a liquid because the thermal energy would be apparent in ways other than just translational movement. The final expressions for real liquids and solids would have several terms, but each would be in units of average energy, just different units.

Again, stuiot alluded to the need for a measurement like T. It takes a lot of the mess out of the calculations. But that’s not what I was arguing. I’m in agreement with T and its application. My assertion was and still is that temperature is a measure of the average energy.