# Why is the total energy of system considered to be a state function? [closed]

Mechanical power is not a state function because it changes with the change of the path taken from a specific initial point to a specific final point.

The same for thermal power. It is not a state function.

But the sum of mechanical and thermal power is the variation of total energy of the system, and the total energy is considered a state function.

I am not convinced since if the mechanical power decreased by some amount, the thermal power doesn't increase by the same amount..

So, why is the variation of the total energy considered a state function?

The argument goes the other way:

• The energy of the system is the sum of the energies of its parts and the interactions between them - this is why it is a state function.
• The energy can be changed by work or transferring heat, which is why their sum is a state function, even though separately neither of them is.

But the sum of mechanical and thermal power is the variation of total energy of the system, and the total energy is considered a state function.

That is true because heat and work are the only two basic means for transferring energy to or from a system. So while neither heat nor work are state functions the combination of the two must necessarily equal the change in the energy of the system (change in state function). For a closed system, the first law is

$$\Delta U=Q-W$$

I am not convinced since if the mechanical power decreased by some amount, the thermal power doesn't increase by the same amount.

I'm not quite sure what you mean since it sounds like you may be confusing "thermal power" with internal energy.

At any rate, it is possible that the work done by or on the system can exactly equal the heat transfer out of or into the system, because in such a case $$\Delta U=0$$. An example is a reversible isothermal expansion or compression of an ideal gas.

Hope this helps.

• Change of the total energy equals to the mechanical power and thermal power. Mechanical power includes change in kinetic energy and Stress Power. Thermal power includes heat sources in the system and heat conduction flux. Power is work per unit of time. So why isn't power directly equal to energy? Why do we say power is equal to energy over time? – user134613 Apr 2 at 14:05
• Why don't we just say that the variation of energy is equal to work? – user134613 Apr 2 at 14:16
• Before I even attempt to address your follow up questions, what do you mean by "stress power".? – Bob D Apr 2 at 15:04
• stress power is the double dot product of the cauchy stress tensor with the strain rate tensor. It is included within the internal energy of a system – user134613 Apr 2 at 16:43
• @user134613 it is a subject of mechanics of materials not the internal energy in thermodynamics. It seems you are mixing up energy transfer with internal energy and mixing up mechanics of materials with thermodynamics. In any event, I don't have the time to sort it out for you. Sorry and good luck. – Bob D Apr 2 at 16:56

I would only add to @Vadim's answer that the internal energy $$E=Q+W$$ being the sum of the absorbed heat $$Q$$ and work $$W$$ is a state function because this is what the experiments are saying going back to at least 200 years from Count Rumford to Joule. It does not matter what $$Q$$ and $$W$$ individually are, if their sum is the same $$Q+W$$ value then the specimen's empirical temperature is the same, i.e., it depends only on $$E$$. True, we do not and can not measure $$E$$ directly but only the empirical temperature, and we also assume that along with the mechanical/electrical/magnetic/chemical parameters a single $$thermal$$ parameter is enough to describe the thermostatic equilibrium state. So far, empirically it appears this is good enough.