system
Given the following post, respond as if you're commenting on Reddit:
human
What is the scientific definition of temperature?

I have been googling this and having a great deal of trouble finding definitions that don't seem circular. In particular, the following is bothering me. Temperature seems to be defined by internal energy (u = 3/2RT) for an ideal gas. Internal energy can be theoretically defined (velocity + vibration + rotation, etc). But specific heat capacity increases with temperature for a real gas, meaning there is more energy per degree of temperature increase. How can there be more energy increase per degree if a degree is literally defined in terms of the amount of energy increase?

Does temperature actually only depend on particle velocity, and other effects just happen to be negligible at normal temperatures? What gives?
gpt
Temperature is defined by:

1/T = (∂S/∂E)*_V,N_*.

That means that the inverse temperature is equal to the partial derivative of the entropy with respect to the energy, with the volume and number of particles left constant.

In other words, it's how much the entropy changes when you change the energy a little bit, leaving the volume and number of particles constant.

This is a totally general definition of temperature, which applies to any system in equilibrium thermodynamics (although technically if you have other extensive quantities in addition to V and N, they should be held constant as well).

This definition becomes equal to 2E/3k for a classical, monatomic ideal gas. This may have been how you defined the temperature in introductory courses, but it's not really a suitable definition for when you're dealing with systems that aren't classical, monatomic ideal gases.