# What is temperature and why is it a fundamental quantity?

1. Sep 29, 2013

### Avichal

Since temperature is just average energy per mole why is it a fundamental quantity ?
Can't we simply have the unit of temperature as Joule / mole ?

2. Sep 29, 2013

### mikeph

Temperature is not average energy per mole, so it couldn't possibly be measured in J/mol.

3. Sep 29, 2013

### Avichal

Then what does temperature represent? I'm convinced that it is not a fundamental quantity and that it can be represented in other units. Why is there a need to have Kelvin as its basic unit?

4. Sep 29, 2013

### stevendaryl

Staff Emeritus
Temperature isn't defined to be average energy, but it is proportional to it for an ideal gas.

The definition of temperature for a system is this

$\dfrac{1}{T} = \dfrac{dS}{dU}$ (with volume and number of particles held constant)

where $S$ is the entropy of the system. You can, as you suggest, choose units in which $S$ is dimensionless (a pure number), and then temperature has the same units as energy. In the usual units, temperature is given a pretty much arbitrary scale, and then this scaling factor is incorporated into the definition of entropy, so that the defining equation for temperature in terms of entropy continues to hold.

5. Sep 30, 2013

### Avichal

Okay, that makes it much clearer. Thank You!
But I still do not understand why give temperature its own scale and not make its unit same as energy. Is it because of convenience i.e the Kelvin and Celsius scales provide more sensible numbers?

6. Sep 30, 2013

### Staff: Mentor

There are most probably historical reasons as to why the SI commitee chose to keep a base unit for temperature.

I remember reading in a book about QFT the author arguing that, just as particle masses are expressed in units of energy, so should temperatures. Personally, I think this would lead to much confusion, especially since temperature is intensive while energy is extensive. For instance, you could have two bodies at temperatures of 100 J and 200 J which would exchange energy until an equilibrium temperature is attained, lets say it is Teq = 175 J. How much energy has been exchanged between the two objects? Except for one special case, the answer will not be 75 J, or 25 J, or 50 J, and will depend on the size of the systems.

7. Sep 30, 2013

### arildno

Remember that not until "heat" was understood to be a FORM of energy in the first decades of the nineteenth century, the concept of "energy", or its predecessor, the "vis viva" was hardly bothered with by physicists. It wasn't regarded as anything fundamental, it wasn't for example, conserved for most systems. With the integration of "heat" into the general concept of "energy" that changed completely.

Temperatures had been regularly measured since the 17th century, so the established scales of temperature were, hardly surprising, not correlated with that of "energy", which one didn't really bother with measuring in the general case. Once it became important to introduce scales for energy proper, it would have been unnatural to seek to make those new units similar to those of temperature.

8. Sep 30, 2013

### SW VandeCarr

Note: You can relate the above definition to an ideal gas as follows:

$$\frac {1}{T} = (\frac {\partial S}{\partial U})_{N,V}= Nk\frac{\partial}{\partial U}lnU^{\frac{3}{2}} = \frac {3}{2} Nk\frac{1}{U}$$

Last edited: Sep 30, 2013
9. Sep 30, 2013

### Avichal

Okay, I am very confused by all this.
Thermodynamics is basically approximation of all the jiggling of atoms and stuff going on the microscopic level.
So heat is the measure of how fast the molecules are moving, volume the space they are occupying, moles the amount of molecules etc.

So what does temperature represent physically? I never understoof temperature, entropy and gibbs free energy properly

10. Sep 30, 2013

### Staff: Mentor

To put it simply: temperature is the tendency of an object to give off heat to another object. Take two isolated systems A and B and put them in contact such that they can exchange energy only in the form of heat, then heat will flow from A to B if and only if $T_\mathrm{A} > T_\mathrm{B}$. In the same conditions, the systems will be said to be in thermal equilibrium if there is no net flow of heat from one to the other, in which case we must have that $T_\mathrm{A} = T_\mathrm{B}$.

The "jiggling" of atoms and all that are just simple representations of what happens to a system when you increase its total energy and hence its temperature relative to some other object.

11. Sep 30, 2013

### HallsofIvy

Staff Emeritus
"Temperature", as you just said, measures the average speed of the molecules, relative to the speed of the center of mass. You seem to be very concerned over whether it is a "fundamental" quantity or not. I'm not sure what you mean by "fundamental" quantity, but it looks like you are talking about having a "unit" of measure (as in "meters") rather than "derived" (as in "meters per second"). Do you not understand that this is a purely arbitrary convention? Many advanced physics papers assume a system of measures in which speed is "fundamental" (because it has a "natural" unit, c) and distance is "derived".

12. Sep 30, 2013

### Avichal

Yes, you are right about my doubt. I don't know why it is bothering me so much but anyways I want to clear it up.

Yes, I think meter is defined in terms of speed of light so distance being a derived quantity. So among distance, time and speed we only need two to be fundamental, other can be derived.
But in case of temperature I feel as if it can derived from other quantities and the fact that it has its own unit and is a fundamental quantity is bothering me.

13. Sep 30, 2013

### D H

Staff Emeritus
No, it's not, for a number of reasons. First off, you should have said kinetic energy, not speed. Helium and argon at the same temperature have rather different average speeds. Kinetic theory presents a nice simple model for an ideal gas (some engineers use the term "perfect gas").

So what's wrong with looking at temperature as energy?

Issue #1: Energy is an extensive property while temperature is an intensive one.

Consider two vessels that each contain helium gas at 300K. One has one mole of helium, the other, two. While the gases in the two vessels have the same temperature, they have different amounts of energy. To make energy intensive you'll have to divide by quantity, e.g., energy per mole or energy per kilogram.

Issue #2: The relation between energy and temperature varies with different substances.

Suppose you have a mole of nitrogen in one vessel, a mole of helium in another, and suppose the two are at the same temperature initially. Now transfer the same amount of heat to both containers. Are the gases at the same temperature? The answer is no. Helium is a monatomic gas, nitrogen is diatomic. Kinetic theory has an explanation for this: Being a diatomic gas, nitrogen has more degrees of freedom than does helium. To look at temperature as energy, you'll not only have to account for quantity, you'll also have to account for different heat capacities.

Issue #3: The relationship between temperature and specific energy (e.g., energy per mole) is not linear.

While understanding kinetic theory is important to gain an understanding of thermodynamics, one also has to realize that it is an ideal model. There is no such thing as an ideal gas (perfect gas). Hold an ideal gas at a constant volume and add a specific amount of energy. The temperature increase will be the same whether that ideal gas is at 100K, 1000K, or 10000K. That's not true for real gases. Consider molecular nitrogen. At room temperature, adding 1.040 kilojoules per kilogram of nitrogen gas raises the temperature of the gas by 1K. At 1000K, you'll need to add 1.167 kJ/kg to get that same temperature change. At 2000K, it takes 1.284 kJ/kg to accomplish the same change.

Issue #4: Kinetic theory only looks at kinetic energy.

In particular, it does not look at potential energy. Consider a mix of ice and water at 0C. Slowly add heat to the mix. Initially the temperature does not change. Some of the ice instead melts. The temperature only starts rising when all of the ice has melted. There are of course ways to model this; it's called heat of fusion.

Issue #5: Looking at temperature as specific energy misses the zeroth law of thermodynamics.

There most certainly is a relation between internal energy, temperature, and entropy. Ultimately, this relationship is a big part of what thermodynamics is all about. That relationship is however nontrivial. Moreover, looking at temperature as specific energy misses something very important: Temperature tells whether heat will flow between two objects in contact with one another, and which way that heat will flow.

14. Sep 30, 2013

### stevendaryl

Staff Emeritus
His suggestion was "energy per mole", which is intensive.

15. Sep 30, 2013

### D H

Staff Emeritus
That's what I said in the paragraph that followed. Even with that, specific energy still does not work as a stand-in for temperature.

16. Sep 30, 2013

### Avichal

Okay, thank you very much. Much of my doubt regarding what is temperature is cleared.

I still have doubt regarding fundamental and derived quantity
I believed that if some physical property can be written in terms of fundamental quantities, it should be derived and cannot have its own unit. So here temperature can be written in terms of joule i.e. 1/T = dS / dU so it should be joule. But turns out that I'm wrong. You can have a fundamental quantity even if it is expressible in terms of fundamental quantities, right?

17. Sep 30, 2013

### SW VandeCarr

I think the problem is different materials respond differently to thermal inputs in terms of changes in temperature. So for equamolar quantities, material A will increase x(a) K while material B will increase x(b) K for 1 J input. This is called the 'specific heat' of a homogenous material. For ideal monoatomic gases, your original definition comes close to being correct, but not for liquids and solids.

EDIT: BTW, "fundamental" dimensional units are somewhat arbitrary. Energy could be a "fundamental" unit and other units defined in terms of energy. i.e $E=ML^2T^{-2}$ becomes $M=ET^2L^{-2}$.

Last edited: Sep 30, 2013
18. Sep 30, 2013

### mikeph

I suspect you're getting hung up on the intuitive link between temperature and energy, because hot objects have more energy. But more massive objects intuitively weigh more, that doesn't mean you should start selling gold by the Newton.

Temperature is not a measure of specific energy, think of it as a variable whose gradient dictates the flow of heat. It is an indicator of the tendency of an object to lose energy to its surroundings. High temperature doesn't always mean high energy. Two objects at exactly the same temperature can have very different specific heats, and two objects with the same amount of energy (per mole or whatever) will not automatically be at the same temperature.

19. Sep 30, 2013

### dauto

Also note that there is indeed arbitrariness in the choice of having temperature while entropy has a derived unit. In fact it is possible to chose entropy as the fundamental quantity and them Temperature will be the derived quantity. It is even possible to define entropy to be unitless and then Temperature would be measured in Joules. All of these are possible choices but for historical reasons Temperature was chosen to have a primitive unit.

20. Sep 30, 2013

### Khashishi

Temperature tells you the direction that energy will spontaneously flow between two bodies. The thermodynamic definition of temperature implies that entropy will increase when energy moves from a hot body to a cold body. Therefore, energy moves from a hot body to a cold body.