# Why total energy equals sum of subenergies

In deriving the chemical potential and also in discussion of intensive properties chemistry books will often do something like this. Say you have a gas and liquid isolated in a container, the total internal energy is equal to the energy of the gas plus the energy of the liquid. Why? I would think they can't be seperated in that way, isn't the liquid exerting electrostatic forces on the gas and vice versa. If you had the same amount of liquid and half the gas surely the internal energy of the liquid would change. What am I not seeing.

I don't believe the internal energy of the liquid would change if you cut the gas in half. Maybe if the container was pressurized and by removing that amount of gas you altered the pressure and as a result changed the liquid's energy, but I'm not so sure otherwise. Ask yourself why we calculate net internal energy? And are the electrostatic forces significant or even thermodynamically relevant?

The internal energy accounts for the kinetic and potential energy of the substance which are functions of temp, pressure, and molecular bonds. It does not account for electrostatic forces or any kind of force fields.

How can the energy of the liquid be independent of the gas and vice versa.Surely each must change the others energy in some way.

Mapes
Homework Helper
Gold Member
The energy of the liquid is dependent only on its state. If that state involves, say, a pressure of 10atm, it doesn't matter if that pressure is being applied by the vapor phase or by a piston. Does this answer your question?

SpectraCat
The statement from your book is correct, but there are some things going on behind the scenes that might help you understand why it is correct:

First, energy is a state function .. that means that it is path independent, so it doesn't matter how a system evolved prior to the instant when you calculate the energy; if you know the instantaneous configuration of the system (kinetic and potential energies of all the particles), then you can calculate the energy. This is what Mapes was getting at. So at any point in time, the total energy of the liquid-gas system in your example is equal to the sum of the energies of the liquid and gas calculated independently.

However, this doesn't mean that there are no interactions between the liquid and gas that need to be taken into account. Suppose for example that the nominal composition of your liquid-gas system is CO2 over pure water in a closed vessel. If you calculated the total energy of the system based on the assumption that the CO2 and water were pure substances, then you would get the wrong answer. This is of course because some of the CO2 dissolves in the water, and some of the water evaporates into the gas phase. If you correct for those effects, then the total energy that you calculate will be lower than for the two-component system consisting of the pure substances.

Does that help?

this helps but as you said the energy depends on the potential and kinetic energies of the particles in the liquid or gas. Wouldn't the particles of the liquid exert forces on the gas. For instance when the partial pressure of a vapor over its liquid is discussed it is said that at the surface of the liquid do to the uneven attraction of all the other particles on those at the surface there is an attractive force, thus when vapor particles get near the surface they are pulled into the liquid, it would thus seem to indicate that when describing the energy of the paricles in the vapor it would be necessary to include this interaction energy.

this helps but as you said the energy depends on the potential and kinetic energies of the particles in the liquid or gas. Wouldn't the particles of the liquid exert forces on the gas. For instance when the partial pressure of a vapor over its liquid is discussed it is said that at the surface of the liquid do to the uneven attraction of all the other particles on those at the surface there is an attractive force, thus when vapor particles get near the surface they are pulled into the liquid, it would thus seem to indicate that when describing the energy of the paricles in the vapor it would be necessary to include this interaction energy.

This is why the earlier poster mentioned that you are not dealing with pure substances - you say "vapor particles get near the surface they are pulled into the liquid," the previous answer said "some of the CO2 dissolves in the water, and some of the water evaporates into the gas phase." Same answer, just different ways of phrasing it (and of course reminding you not to forget the water going into the gas phase!).