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

• Avichal

#### 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 ?

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

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

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 ?

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.

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.

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?

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?
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, let's 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.

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.

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

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

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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

So what does temperature represent physically?
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.

"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".

"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").
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.

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".
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.

"Temperature", as you just said, measures the average speed of the molecules, relative to the speed of the center of mass.
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.

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.

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

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.

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?

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?

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 $M=ET^2L^{-2}$.

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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.

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.

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.

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.
That is true, but it does not really explain why that occurs.

Heat flows spontaneously from a body at higher temperature to a body at lower temperature because the average translational kinetic energy of the molecules in the hotter body is greater than the average translational kinetic energy of the molecules in the cooler one. This is true not only for ideal gases but for all matter. (It is only not quite true when you get into very low temperatures near absolute zero when the concept of translational kinetic energy becomes unclear).

The underlying physical principle here is that matter tends toward a state of thermal equilibrium. This is just a statistical law. When I send a high speed cue ball down a table with 15 stationary billiard balls, the result inevitably is that the cue ball's initial energy will end up being distributed among all 16 balls.

If I had 15 trillion slowly moving billiard balls and fired 1 trillion high speed cue balls into them, the tendency would be for the speeds of the cue balls to decrease and that of the initially slower billiard balls to increase. The second law of thermodynamics says that this must occur-that a scenario in which the slower balls will randomly collide with the cue balls to increase the average energy of the cue balls is statistically it is so unlikely that it will never occur in the real world.

So in order to study heat flow, we need to measure of the average translational kinetic energy of the molecules of a substance. This measure is called temperature. And the statistical law is called the second law of thermodynamics.

AM

The thing is, you can define temperature even in a system with no translational kinetic energy. The concept of temperature is more general than any particular type of energy. The only thing that is needed is some kind of energy transfer and a density of states.

Things don't have to physically touch to transfer energy. If you have a hot sun and a cold planet, then heat moves from the sun to the planet (through the emission of photons). The planet is hotter than the the background space, so the planet radiates heat into space. It doesn't matter how two bodies are connected--perhaps physically touching, or radiatively coupling, or magnetically coupling--the direction of energy flow is always from hot to cold. The mechanism only changes the rate. "Why" energy goes from hot to cold isn't really that important unless you want the rate.

Temperature is a thermodynamic concept, and it is essentially independent of whatever underlying kinematic theory is being used. There was a quote from a famous physicist that said something like, all our models in physics will probably be superseded with more accurate theories, except for thermodynamics, which is correct.

An example of a scenario without kinetic energy is spin temperature. That is the temperature of a system where the only important interaction is between the spin of some particles and an external magnetic field.

That is true, but it does not really explain why that occurs.

Heat flows spontaneously from a body at higher temperature to a body at lower temperature because the average translational kinetic energy of the molecules in the hotter body is greater than the average translational kinetic energy of the molecules in the cooler one. This is true not only for ideal gases but for all matter.

No, that isn't completely true. The statistical mechanics approach to thermodynamics explains thermal equilibrium this way:

Suppose that we have two systems $A$, and $B$, having total energy $E$. The two systems are in thermal contact (that is, they can exchange energy, but not particles). Because of thermal randomness, energy will move back and forth between the two systems, and the division of the total energy $E$ between the two system will fluctuate. If we let $E_1$ be the energy of $A$ and $E_2$ be the energy of $B$, what is the most likely value for $E_1$ and $E_2$?

Well, we can make a counting argument: Let $W_1(E_1)$ be the number of possible ways to arrange $A$ so that it has energy $E_1$. Let $W_2(E_2)$ be the number of possible ways to arrange $B$ so that it has energy $E_2$. Then the total number of ways to arrange the composite system so that it has total energy $E = E_1 + E_2$ is given by:

$W(E) = W_1(E_1) \times W_2(E - E_1)$

The most likely value for $E_1$ is assumed to be the value that makes $W(E)$ largest. So to find $E_1$, we want to maximize the function:

$f(E_1) = W_1(E_1) \times W_2(E - E_1)$

where I used the fact that the total energy is $E$ to rewrite $E_2 = E - E_1$.

To maximize a function, you take the derivative and set it to zero. So we have, for a maximum:

$\dfrac{dW_1}{dE_1} W_2 - \dfrac{dW_2}{dE_2} W_1 = 0$

We can rearrange this as:

$\dfrac{\frac{dW_1}{dE_1}}{W_1} = \dfrac{\frac{dW_2}{dE_2}}{W_2}$

which can be written, using properties of the logarithm, as:

$\dfrac{d (ln(W_1))}{dE_1} = \dfrac{d (ln(W_2))}{dE_2}$

The connection between these formulas and thermodynamics is to make the associations:

• The entropy of system $A$ is a function of its energy $E_1$: $S_1(E_1) = k ln(W_1(E_1))$ where $k$ is Boltzmann's constant.
• The temperature of system $A$ is defined to be: $T_1 = \dfrac{1}{\frac{dS_1}{dE_1}}$
• The entropy of system $B$ is a function of its energy $E_2$: $S_2(E_2) = k ln(W_2(E_2))$.
• The temperature of system $B$ is defined to be: $T_2 = \dfrac{1}{\frac{dS_2}{dE_2}}$
• The systems are said to be "in thermal equilibrium" if $E_1$ and$E_2$ are at their most likely values, which is when $T_1 = T_2$.

This definition of temperature and entropy does not assume anything about the nature of the two systems. But if you do assume that the two systems are made up of particles, and that all the energy is kinetic energy of motion, then that allows you to calculate $W_1$ and $W_2$, and you find that the temperature is proportional to the average kinetic energy. So you don't have to assum that.

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.
Right.
Think about a rock and a block of ice and the effect of a phase change. The ability to transfer energy between them can be understood easily it terms of temperature, but what other units allow that fundamental understanding and is consistent through phase change?.

It is only convention (and history) that we have temperature as one of the 7 base SI dimensions. You are absolutely right that temperature can be derived from the internal energy and entropy. As others have said, temperature could have been replaced with another derived quantity, but temperature is pretty easy to measure.

In the bigger picture, there is nothing magic about needing 7 base dimensions in the SI system. One could argue that there are only two base dimensions - distance and time - and all others are derived from them. I even seen arguments that only distance needs to be a base dimension.