# What underlying physical quantity do we measure when measuring the temperature of gases?

When we measure the temperature of, say air, or any other gas, what physical quantity do we measure? Is it the kinetic energy of molecules? Is this affected by pressure and density? And if so, how?

• The temperature is defined as the inverse of the derivative of entropy w.r.t. energy. Of course, in reality you measure the temperature by bringing the system in thermal contact of a small system that we understand, and whose properties we can measure, like the volume of some liquid in a thermometer.
– user178876
Sep 22, 2018 at 22:34

The assumption that there is a particular underlying physical quantity that we are measuring when we measure temperature is not necessarily correct. Temperature in itself is a physical quantity, and in particular is one of the fundamental properties of systems of many particles. It can be defined in the following way: temperature is how much energy the system gains if, while holding the other macroscopic variables constant, you increase the number of states the system has access to (which is known as the entropy). Specifically, for a system with internal energy $$U$$ and entropy $$S$$, temperature is defined as

$$T=\frac{\partial U}{\partial S}$$

You could say that a measurement of temperature is measuring how the system's internal energy responds to a change in the number of accessible states. You could also say that a measurement of (inverse) temperature measures how the number of states accessible to the system changes with energy.

In general, temperature is related to other thermodynamic variables by an equation of state. The specific form of the equation, as well as the variables involved, depend highly on the specific properties of the system. For a monatomic ideal gas with pressure $$P$$, volume $$V$$, and number of particles $$N$$, this is the ideal gas law, $$PV=Nk_BT$$. For a real (non-ideal) gas, this is the van der Waals equation, $$\left(P+\frac{Na}{V}\right)(V-b)=k_BT$$, where $$a$$ and $$b$$ are empirically-measured constants. For liquids, one possible equation of state is the Tait equation. For degenerate gases (found in the cores of massive stars), the equation of state is $$P=k(N/V)^{4/3}$$, where $$k$$ is an empirically-measured constant.*

That said, there is one ideal system for which there is an unambiguous link between the macroscopic variable of temperature and the microscopic behavior of individual particles in a system: the monatomic ideal gas. According to kinetic theory, temperature is a measurement of the average kinetic energy of an individual particle; namely, in three dimensions, $$T=\frac{3}{2}k_B K_{avg}$$, where $$k_B$$ is Boltzmann's constant and $$K_{avg}$$ is the average kinetic energy. The temperature of a monatomic ideal gas is related to its pressure $$P$$ and density $$\rho$$ by the ideal gas law, $$P=\frac{\rho}{m}k_B T$$, for particles of mass $$m$$.

Let me stress, though, that this relation only holds for monatomic ideal gases. For ideal gases made of molecules, there is an ambiguity in whether you include the rotation and vibration of the molecule as "kinetic" energies, and as such there may be different definitions of temperature (not necessarily equivalent to the first one I described) to cover different measurement situations.

For real (non-ideal) gases, too, the presence of interactions means that the way the system's entropy responds to energy input (i.e. the temperature) can change without the average kinetic energy changing, if the added energy goes into increasing the potential energy of the system.

The further from a monatomic ideal gas you get, the less the kinetic energy of an individual particle and the temperature of the system are related. In fact, there are systems in which trying to make this kind of link makes no sense whatsoever! For example, a chain of magnetic particles subject to an external magnetic field has a temperature based on the response of the orientation of the magnetic axis of those particles to an increase in energy, even though the particles are not moving, and there is no energy in the system that could be called "kinetic energy." In addition, there are systems with a very limited state space which can take on negative temperatures, while kinetic energies can only take on positive values. This can be explained by the fact that, in this system, temperature has nothing to do with kinetic energy - due to the limited state space, the system's number of available states decreases as energy is added (you can picture a system that has a maximum energy per particle, where adding energy just packs more and more particles into that maximum energy state). So, in general, there is no single underlying microscopic physical quantity that is related to temperature.

*Note that this equation of state does not depend on temperature, meaning that the temperature of a relativistic degenerate gas can increase without affecting the pressure or density. This is one of the reasons that the end of a star's life is often so violent - the core of the star becomes degenerate, its density becomes essentially fixed at the maximum possible value, and no matter how hot the core gets, it cannot produce enough pressure to hold off the matter above it. As such, the star collapses.

• What do you mean by "magnetic particles" ? Sep 29, 2018 at 1:25
• @N.Steinle I was attempting to refer to the Ising model without having to actually go into details. The "magnetic particles" could be the classical analogue of atomic magnetic moments, or they could be grains of a paramagnetic substance that interact in the same way, hence the general term "magnetic particles." Sep 29, 2018 at 16:02
• I see. So, by magnetic particle you mean "any particle with a well defined magnetic moment" ? Sep 29, 2018 at 17:06

The equation $pV=nRT$ is a well known one for “ideal gases” and whilst in reality no gases behave like this, it does demonstrate the kind of link. If we look at the absolute temperature scale we see it must rely some way on energy, otherwise why would there be a minimum? I believe there is also an equation $\frac{1}{2}mv^2=\frac{1}{2}nkT$ which demonstrates this link between energy, velocity and temperature nicely.

• There's only a minimum temperature in classical thermodynamics, so this argument doesn't hold in general, as evidenced by the existence of negative-temperature systems. And your second equation is also only valid for (monatomic) ideal gases. Sep 23, 2018 at 0:02

When we measure the temperature of air or any other gas we are measuring the average translational kinetic energy of the molecules. This is sometimes referred to as the "kinetic temperature".

If the gas approximates ideal gas behavior, that is if molecules are far enough apart so that intermolecular forces can be neglected, then the relationship between temperature, specific volume (the inverse of density) and pressure follows the ideal gas law. Or

$$Pv=RT$$

Where P = absolute pressure (Pa)

v = specific volume ($$\frac {m^3}{kg}$$) (the inverse of density)

R= specific gas constant

T = absolute temperature (deg K)

Hope this helps

• Your equation for the ideal gas law is missing the molar number, $n$, on the RHS. Sep 22, 2018 at 22:47
• No it isn't. You need moles, n, if you use total volume, V, (cubic meters) instead of specific volume, v, (cubic meters per kg) Sep 22, 2018 at 23:02
• Ah i see that now, sorry missed the word "specific." Sep 22, 2018 at 23:03
• @N.Steinle. But I did mistakenly say universal gas constant. It should have been the specific gas constant. I made the correction. Thanks. Sep 28, 2018 at 19:12