# Temperature: Why a Fundamental Quantity?

Temperature is just an indication of a combined property of the masses of the molecules and their random motion. In principle, we can explain "no effective energy transfer between two conducting solid bodies in contact" via a condition in terms of the masses of the molecules and their speeds such that due to the collisions of molecules of two bodies, net energy transfer between two bodies is zero. But it would be a complex calculative work to derive this condition analytically so we use the temperature scale just as a phenomenological parameter to easily determine the condition of "no net energy transfer between conducting solids" for practical purposes. But it does not denote any fundamentally new property of the body separate from the already known mechanical properties of the same. Then why do we call it a fundamental quantity, e.g. in the SI list of fundamental quantities?

• It's exactly that - it's easier, simpler, as well as easier to explain temperature. On a macro scale, it does appear fairly fundamental. Commented Feb 2, 2014 at 18:10
• on a micro scale temperature does in fact also have an (abstract meaning) as $T=\frac{1}{k_B}\left(\frac{\partial \ln (\Omega)}{\partial E}\right)^{-1}$. This is a definition used in statistical physics. Commented Feb 2, 2014 at 18:13
• What is your physics level? The equation Hagadol posted means that temperature is the variation of energy with respect to the number of microscopical states. Commented Feb 2, 2014 at 18:24
• There is no such thing as an "SI list of fundamental quantities"; the concept you refer to is that of a base unit, and it is important to use the correct term. This question should be edited to use correct language. Commented Aug 16, 2017 at 18:20
• Temperature is an SI unit. It is not a fundamental quantity in the sense that you mean and is not called that in the SI system (joules, in contrast, is a more fundamental concept). Lots of units are not "fundamental" and are not measurements of something fundamental (see also, e.g., moles). Commented Feb 12, 2019 at 8:51

It is one of the fundamental questions in classical thermodynamics.

Temperature: Temperature is the parameter that tells us the most probable distribution of populations of molecules over the available states of a system at equilibrium.

We know from Boltzmann distribution:

$$\beta=\frac{1}{k_B T}$$

The fact is that $$\beta$$ is a more natural parameter for expressing temperature than T itself.

Absolute zero of temperature (T = 0) is unattainable in a finite number of steps, which may be puzzling, it is far less surprising that an infinite value of (the value of $$\beta$$ ‚ when T = 0) is unattainable in a finite number of steps. However, although $$\beta$$ is the more natural way of expressing temperatures, it is ill-suited to everyday use.

The existence and value of the fundamental constant $$k_B$$ is simply a consequence of our insisting on using a conventional scale of temperature rather than the truly fundamental scale based on $$\beta$$. The Fahrenheit, Celsius, and Kelvin scales are misguided: the reciprocal of temperature, essentially $$\beta$$, is more meaningful, more natural, as a measure of temperature. There is no hope, though, that it will ever be accepted, for history and the potency of simple numbers, like 0 and 100, and even 32 and 212, are too deeply embedded in our culture, and just too convenient for everyday use.

Although Boltzmann’s constant $$k_B$$ is commonly listed as a fundamental constant, it is actually only a recovery from a historical mistake. If Ludwig Boltzmann had done his work before Fahrenheit and Celsius had done theirs, then it would have been seen that ‚ was the natural measure of temperature, and we might have become used to expressing temperatures in the units of inverse joules with warmer systems at low values of ‚ and cooler systems at high values. However, conventions had become established, with warmer systems at higher temperatures than cooler systems, and k was introduced, through $$\beta=\frac{1}{k_B T}$$, to align the natural scale of temperature based on ‚ to the conventional and deeply ingrained one based on T. Thus, Boltzmann’s constant is nothing but a conversion factor between a well-established conventional scale and the one that, with hindsight, society might have adopted. Had it adopted ‚ as its measure of temperature, Boltzmann’s constant would not have been necessary.

Conclusion: Temperature, actually is NOT a fundamental quantity. It is just for convenience and historical reasons we consider it as fundamental quantity.

Reference: Peter Atkins -The Laws of Thermodynamics: A Very Short Introduction

As I already commented one can introduce the Temperature of a gas by relatively modest assumptions. Here is a sketch of a derivation I hope to remember correctly:
The definition of temperature is then based on the concept that if two gases are brought together the entropy will maximize. This condition can be simplified to the condition, that the two inverse "temperatures" have to be the same. This yields the formula I already gave, namely $$\frac{1}{k_B T}=\beta= \frac{\partial \ln (\Omega)}{ \partial E}.$$ Here, $k_B$ is a scaling constant,$E$ is the energy and $\Omega$ something like the number of available states for the system with a given energy.
For a proper derivation you can have a look in practically every book on statistical physics.

So why do we say it a fundamental quantity?

You do not have to say such thing, but temperature is very basic and important concept. In thermodynamics, it is the only quantity that always gets equalized in transition to thermodynamic equilibrium - pressure nor chemical potential needs to equalize, but temperature has to (except perhaps for systems in strong gravitational field, where the lower parts are predicted to have higher temperature than the upper parts).

"Temperature is just an indication of a combined property of the masses of the molecules and their random motion."

No! Temperature is not always limited to being a combined property of the masses of the molecules and their motion. Of course, this was the first scenario where the notion of temperature became apparent to the humans historically, but our modern notion of temperature transcends this primitive notion of temperature as being some kind of a measure of the kinetic energy of molecules. Rather, the temperature is a quantity that generically represents whether a given system will be in an equilibrium when kept in contact with another system. More specifically, it represents whether the two systems can exchange energy with each other and attain a combined final state with more number of compatible microstates than the number of microstates compatible with the combined initial state of two systems. If they can then they would evolve to that state and otherwise not. In fact, we don't even postulate that such a quantity must exist but it follows from the basic postulates of statistical physics that such a quantity would exist and then we identify this statistical quantity with the thermodynamic temperature to make a contact between our theoretical framework and the experimental results---as needed to be done in case of any theoretical framework.

Now, the key is that this statistical quantity that we identify with the thermodynamic temperature is rather general and conforms to our primitive notion that ''the temperature has to do with kinetic energies of molecules'' only if the Hamiltonian of the system is that of a classical ideal gas, $H=\displaystyle\Sigma_{i}\dfrac{p_i^2}{2m}$. There certainly exists very many Hamiltonians (that is to say, very many systems) where there might be many other terms in the Hamiltonian which do not represent the kinetic energy of molecules and there might not even be any notion of the motion of molecules at all (e.g., there are no kinetic energy terms in the Hamiltonians representing magnets etc.--and yet, the concept of temperature as a quantity defined in the statistical sense we discussed makes perfect sense on its own!) So, in short, the text in the blockquote is not really true in the light of our modern understanding of temperature.

Now, as to whether the temperature is a fundamental quantity or not, as clearly evident from the statistical definition of temperature, the quantity called temperature emerges from the more basic statistical considerations and is not thus fundamental in the sense that it is not irreducible to more basic notions. But certainly, it is a very important quantity for both theoretical and experimental purposes and can be regarded as fundamental in this sense. Whether it absolutely requires a unit of its own has a definite negative answer. But again, from a theoretical point of view, every quantity can be expressed in terms of just one unit, say $eV$---but clearly, it wouldn't be convenient and thus, it is certainly wise to use a separate unit for temperature (and other quantities as well) despite the fact that we can use a more unified framework of units.

• "But again, from a theoretical point of view, every quantity can be expressed in terms of just one unit, say eV" We can actually go even further than that, fix G = 1 and express even energies as dimensionless quantities with the implied meaning as multiples of the Planck mass, just like dimensionless velocities are fractions of c and dimensionless actions are multiples of h Commented Jun 7, 2023 at 10:30

A quantity is called as fundamental quantity if it can't be explained in terms of other fundamental quantities:

• we know temperature is the vibrations and collision of constituent atoms and molecules and
• vibration can be explained by other known fundamental quantities.

Hence temperature is not a fundamental quantity.

But wait... Kelvin is a fundamental unit!

In the past, temperature was used for the measurement of "hotness". For that we (Humans) devised different temperature scale and laws like the Zeroth Law of Thermodynamics.

Then as we encountered more physical phenomenon such as thermodynamic equilibrium, we found that this quantity, the temperature, is the same for two systems in equilibrium.

At that time we usually dealt in macroscopic domains but as we started to research microscopic domains, we can explain temperature as the vibrations and collision of molecules and atoms.

It's a lot easier to measure temperature than to measure the motion of component particles. Hence, we can accept it as a fundamental quantity.

It is just a scale to get thermal equlibrium problems easily but defined in such a way that it can not be expressed only in the terms of the other fundamental quantities.So it is a fundamental quantity.

Just as an adder to previous answers: in thermodynamics, you most often see temperature associated with R (or k in statistical thermo), i.e. most formulas have RT or kT as a parameter. The combination RT is thermodynamic temperature and the units are J/mol, so no need for another fundamental unit for temperature. Temperature can be fundamentally defined as length squared over time squared.

But it would be a complex calculative work to derive this condition analytically so we use the temperature scale just as a phenomenological parameter to easily determine the condition of "no net energy transfer between conducting solids" for practical purposes.

I'd say temperature is just a general "tool". Actually all concepts defined by statistical physics are such "tools". All properties of a system derive from its Hamiltonian and the underlying mechanical laws. Statistical physics introduces general concepts to bypass working with the explicit equations of motion, ideas that work for all Hamiltonians and all underlying interactions.

As mentioned by @youpilat13, the temperature defined in statistical physics is more general than the idea of kinetic energy per participle (more exactly per degree of freedom). The temperature is equal to the kinetic energy per degree of freedom essentially if the Hamiltonian has the form: $$H=U(q)+\sum \frac{p_i^2}{2m_i}$$ where $$U$$ is a scalar potential. See: Statistical mechanics definition of temperature as the average kinetic energy. This is a consequence of the equipartition theorem.

The temperature is defined more generally as: $$T=\left(\frac{dE}{dS}\right)_V$$

$$V$$ is not always the real volume, it can be used to refer to any external intervention. The second law implies that the temperature defines the direction of energy in absence of external intervention: heat flows towards the body with lower temperature. It is easy to see. Two bodies $$A$$ and $$B$$ in thermal contact but thermally isolated from the rest of the universe are such that: $$\frac{\delta Q}{T_A} - \frac{\delta Q}{T_B}\geq 0$$

This idea is general. Whenever a variable $$X$$ is both extensive and conservative like the energy $$E$$, you can define: $$T_X=\left(\frac{dX}{dS}\right)_V$$

And then, $$X$$ will flow in the direction of decreasing $$T_X$$. An example is $$X$$="the number of particles of hydrogen". All this to say that what is more fundamental is $$S$$, and $$T$$ is somehow a little tool used about energy to predict the direction of exchange as it could be done for any other conservative extensive quantity.