# Why is internal energy a function of temperature and volume?

Why is internal energy a function of temperature and volume? $u=f(T,V)$. In my reasoning, I can only understand that internal energy is a function of Temperature, because it changes directly with Temperature. If temperature raises, $u$ raises, if $T$ decreases, $u$ decreases.

What about enthalpy? Why is it a function of temperature and pressure? $h=f(T,p)$

• That may be true for an ideal gas but it's not true in general... Commented Apr 22, 2017 at 16:32
• From the phase rule, it takes two independent degrees of freedom to specify the thermodynamic equilibrium state of a single phase, single component substance. Commented Apr 22, 2017 at 19:12
• It may be worth looking up 'state variables' and 'extensivity'.
– user140374
Commented Apr 23, 2017 at 8:43

## 2 Answers

Internal energy includes both the kinetic energy and the potential energy (of interaction) of the molecules. As the specific volume gets smaller, the molecules squeeze together more, and the potential energy (and thus the internal energy) changes. As the molecules get further apart, the potential energy makes less and less of a contribution, and the internal energy is determined solely by the temperature (kinetic energy).

When the temperature of any system rises, the kinetic energies of the particles also increase causing the internal energy of the system to rise or vice-versa. For any physical state other than gas, volume is directly proportional to the no. of molecules. So if the volume increases, the total no of molecules also increase resulting in the rise of internal energy or vice-versa. In this way, the internal energy of a system is dependent on temperature as well as volume.