>> Here we have a situation where we start out with some mass of a liquid,
in this case it's called water at some temperature, T1 and there's a piston
above the liquid and it's held in place by a stop, and there's vacuum above it,
and what we're going to do is pullout the stop, let it go to equilibrium.
So, what's going to happen is some of the water is going to evaporate and we'll call--
we'll call the mass that evaporates, mass 2 vapor,
so the mass of liquid would be the initial mass minus how much evaporates.
The temperature is going to be different than, of course,
now we're going to have some pressure whereas we started with no pressure above the piston.
And so, we would like to determine how much evaporates and what is the temperature
at equilibrium and corresponding what is the pressure?
And so, we can look at this system to help visualize what's happening.
Let's say a small amount evaporates.
Well, it needs to have energy to evaporate.
That comes from cooling down everything and most of that is the liquid, because there is going
to be much more liquid than vapor volume.
And so, we have to cool down the system.
We're assuming the heat capacity of the container is 0 to make the calculations easier.
We're assuming it's 80 [inaudible] it.
And if a small amount evaporates, the temperature drops,
the amount that evaporates then we can determine the pressure
from the volume amount evaporated ideal-gas law molecular weight, and then we can check does
that correspond to vapor liquid equilibrium?
This is the pressure, the saturation pressure.
If not, let's look at a little more evaporates; we cool it down a little more,
eventually we're going to be at equilibrium where the pressure, the saturation pressure
and the temperature of course saturation temperature.
So, the equations that we're going to use we have to one,
and that's going to be the ideal-gas law, that relates the volume of the container
and the final pressure to the mass in the vapor phase and we'll divide it
by its molecular weight, and the gas constant and the temperature.
Well, we need to have vapor liquid equilibrium and vapor liquid equilibrium assumes
that we're at saturation pressure.
So, we need an equation and we're going to just write down Antoine equation,
because that makes it easiest to solve, for that should be temperature T2, so T2 plus C, A,
B and C are all constants and then the last equation we need is the energy balance.
It's a closed system.
The change in internal energy, the total not per mole, but the total is going to be 0.
And so, we want to write down the equation that says the initial amount, so we only have liquid,
so keep in mind this is per gram or kilogram, but per mass times the mass,
that's how much the initial internal energy.
The final is, well we have liquid remaining which is mass 1 minus how much evaporated,
total energy of liquid final, and then we have the mass
that evaporated times internal energy of the vapor.
So, that's the energy balance for this system and what we want to do is then substitute
in so we can write this in terms of heat capacities and Delta U of vaporization.
Now, remember whenever we work with internal energy we're always looking at changes
since there's no absolute values, so we can pick a reference system.
So, so for example, here's what we're going to do.
We can say internal energy of a liquid at some temperature is internal energy
at the reference temperature plus the heat capacity of the liquid;
temperature minus a reference temperature.
Now, keep in mind for a liquid this is essentially the same as Cp.
It's not true for a gas, but for a liquid and we can make our life easy
by saying our reference value is going to be 0 at T reference equal
to T1 our starting temperature, okay?
So, let's make this substitution then into our energy balance.
So, for the left side of the equation, I changed it to Cp for the liquid,
the temperature which starts T1, our reference temperature is T1, so of course this is going
to end up being 0 for the term on the left side.
The right side let's look at first the amount of liquid.
So, the amount of liquid since, I'm going to use heat capacity for the liquid,
T2 minus our reference temperature T1.
And now for the vapor, it's a little more complicated.
There is the mass of the vapor so again, we can look at this in different ways.
What we're going to do is we can visualize the vapor--
we can visualize the vapor evaporating at our reference temperature and then heating
up that vapor to the final temperature.
So, at our reference temperature for liquid, the internal energy is 0
and then we have a change due to vaporization, and we'll use the notation at temperature T1
and then we have Cv for vapor, T2 minus T1 our reference temperature.
Well, this is our energy balance.
So, remember this is 0, so we have an equation where our T2, mass of vapor, and then of course,
we have 3 equations, so P2, so we have 3 [inaudible]
and then the 3 equations are our ideal-gas law, Antoine's equation, and Delta U total equal 0
which we just had written in terms of masses.
So, we have 3 equations that we can solve simultaneously
to determine the final conditions, how much evaporates and what's that final temp.