# How to define exactly the internal energy?

I would like to understand the first law of thermodynamics, but I have some problems with the basic concept. How to define exactly the internal energy?

As I see it, given a system of particles, which the principle of work is in force for. Can it be considered 'per definitionem' the isolated thermodynamical system? If yes, the diatermic system can be defined as a system where the priciple of work isn't met. Therefore, if we define the internal energy as the sum of kinetic and potential energies of the particles, there is a term missing from the equation of the sum-work in diatermic case, called heat, which the first law postulates. How can heat be mathematically characterized?

• What is "principle of work"? – Deep Oct 21 '16 at 11:10
• Principle of work (and energy) is that the sum of elementary nonconservative works equals to the change of the mechanical energy. – TobiR Oct 21 '16 at 11:13
• Hi TobiR, I removed your last subquestion, cf. this meta post. – Qmechanic Oct 21 '16 at 12:58
• This article may be helpful: physicskey.com/47/…. – Saral Sep 10 '18 at 2:47

The first law tells you that you can change the internal energy of a system $\Delta U$ by either having work done by the system $W$ or adding heat to the system $Q$

$$\Delta U = Q-W$$

The system does not contain any heat.
Heat and work are not state functions of a system.
So you cannot say that this system has so much heat in it at one time and more heat in it at another time.
However you will see $\delta Q$ in textbooks which means a small amount of heat added not a change in the amount of heat in the system.
Another common form which I cannot reproduce here is a little $d$ with a line through it.

The internal energy of a system is the sum of the kinetic energies and the potential energies.

If a system does no work and you add heat to it, the internal energy of the system increases.
If your system is an ideal gas then this increase in internal energy is an increase in the kinetic energy of the atoms of the gas.
For most systems heat entering a system will affect both the kinetic energy and the potential energy of the system.

Temperature can be defined for a system in equilibrium and so you can define a difference in temperature.

• From what can it be derived that U is a totally differentiable function? U is similar to the "total mechanic energy"? – TobiR Oct 21 '16 at 10:57
• What is the heat? Is it a discrepance compared to the principle of work? – TobiR Oct 21 '16 at 10:59
• Temperature is a consequence of the zeroth law of thermodynamics, but I don't know, how to derive that T(V,p) is a totally differentiable function with respect to V and p? – TobiR Oct 21 '16 at 11:02
• Could you help me in what is the exact definition of the thermal equilibrium without the notion of heat? – TobiR Oct 21 '16 at 11:04
• In principle you can find values of the internal energy of a system and so you can find the change in internal energy. Heat is that form of energy which is transferred from a region at a high temperature to a region which is at a lower temperature. Two bodies are in thermal equilibrium if no heat is being transferred between them. On the molecular scale it means that if atoms from the bodies interact there is no net transfer of kinetic energy between the atoms. What do you understand by differentiability? – Farcher Oct 21 '16 at 11:19