Alright, let's take a look at another problem from a recent exam. Here we have two different
kinds of systems that are initially at the same temperature, pressure and composition
conditions, and we're going to add 100 kilojoules of heat to these two systems - I'm calling them System
1 and System 2. Alright, System 1 is a constant volume system, so it has fixed boundaries.
We'll add 100 kilojoules of heat to it. Right, System 2 is a constant pressure system, so
we have a piston-cylinder that will expand after we add heat to it. And we want to basically
answer the question of whether T1 is going to be greater than, less than or equal to
the final temperature for the second system. Alright, so to answer this we'll need to do
an energy balance for each system. So the First Law for a closed system like each of these
systems where no mass is crossing the boundaries of the system, is delta U is equal to Q plus
W. Alright, we know that in System 1, that no work is done, there are no moving parts
in the system. There's no expansion or contraction of the system because it's constant volume.
And so the work is zero, and therefore delta U sub 1 is equal to this 100 kilojoules of heat
that we've added. Alright, so that heat entirely goes to raising the internal energy of the
system. Alright, if we look at System 2, has the same basic form of the energy balance,
but in this case, as we add energy to the system we're going to expect that we can actually
push up on the cylinder. Alright, as we increase that internal energy, some of that will actually
be transferred to the surroundings in the form of doing work on the surroundings. Alright,
and so, because we'll push up on this cylinder, then the system is doing work on the surroundings,
and that's taking away from the internal energy. It takes energy to do work. And so therefore this work term is
negative because of the expansion in volume. This term is still 100 kilojoules. Alright, and so
the bottom lines is that delta U sub 2 will be some number less than 100 kilojoules. Alright,
and so we know that the internal energy goes up by less in System 2 than in System 1. And
so therefore, in the final state, U sub 1 is going to be greater than U sub 2. And the
question is: how does this correlate with temperature? OK if this was an ideal gas,
alright then U is only a function of temperature and not of pressure. Alright, and therefore
if you had a higher internal energy in tank 1 than in tank 2, you would also have a higher
temperature in tank 1 than tank 2. Because U is directly proportional to temperature.
Alright, in this case though, we're given that it's a non-ideal gas, OK, and the question
is whether that is going to affect our conclusion that T1 is greater than T2. Alright, and it
turns out that it doesn't affect that. Alright, U can be a function of both temperature and
pressure. Alright, in this case, we expect, when we add the energy to tank 1 at constant
volume, that's going to have the capability to increase the pressure. Alright, it might
not increase it in a directly proportional way like we would expect for the ideal gas
law when we increase that temperature, but as we add energy to the system, we're going
to expect the pressure to go up in a constant volume system. Alright, in the case of tank
2, the system can expand and actually the pressure won't go up at all, it will be the
same pressure that we started with. And so therefore P1 is going to be greater than P2.
Now, typically, the internal energy - which for a non-ideal gas can be a function of pressure
- as the pressure goes up, then typically the internal energy - if we're operating at
fixed temperature - will go down. So gases at higher pressures but the same temperature
have a lower internal energy. Alright, you can see this if you look up data, for example,
in the steam tables. And the reason for this is that gases at higher pressure interact
with each other more frequently. Alright, and when they experience these interactions,
those interactions are actually attractive, and they stabilize the system. Alright, so
gases will tend to interact with each other to lower the internal energy of the system.
Alright, and so we would expect, based on pressure effects alone, for the internal energy
of the system to be lower. But that's not the case, we know already that the internal energy
of System 1 is greater than the internal energy of System 2. And to compensate for that, the
temperature must be higher, because as we increase temperature, we increase internal
energy. And so, it actually is not the case that if we have a non-deal gas that that ultimately
affects our conclusion; we still find that T1 is greater than T2. And I should point
out that for most gases under most engineering applications that we look at, the temperature
effects on internal energy will be much, much greater than the pressure effects on internal
energy.