Page 46 - Physical chemistry eng

2.4 DOING WORK ON THE SYSTEM AND CHANGING THE SYSTEM ENERGY FROM A MOLECULAR LEVEL PERSPECTIVE 23

effect the transformation. The densities of liquid and gaseous water under these condi-
–3
tions are 997 and 0.590 kg m , respectively.
a. It is often useful to replace a real process by a model that exhibits the important
features of the process. Design a model system and surroundings, like those
shown in Figures 2.1 and 2.2, that would allow you to measure the heat and work
associated with this transformation. For the model system, define the system and
surroundings as well as the boundary between them.
b. How can you define the system for the open insulated beaker on the laboratory
bench such that the work is properly described?
c. Calculate q and w for the process.

Solution
a. The model system is shown in the following figure. The cylinder walls and the
piston form adiabatic walls. The external pressure is held constant by a suit-
able weight.
b. Define the system as the liquid in the beaker and the volume containing only
O in the gas phase. This volume will consist of disconnected
molecules of H 2
volume elements dispersed in the air above the laboratory bench.
c. In the system shown, the heat input to the liquid water can be equated with the
P external 1.00 atm
work done on the heating coil. Therefore,
3
q = Ift = 2.00 A * 12.0 V * 1.00 * 10 s = 24.0kJ
As the liquid is vaporized, the volume of the system increases at a constant exter- Mass
nal pressure. Therefore, the work done by the system on the surroundings is
5
w =-P external (V - V ) =-10 Pa Heating coil
i
f
-3
10.0 * 10 kg 90.0 * 10 -3 kg 100.0 * 10 -3 kg H 2 O (g)
* ¢ + - ≤ A
0.590kgm -3 997kgm -3 997kgm -3 12 V H O (l)
2
=-1.70kJ
Note that the electrical work done on the heating coil is much larger than the P–V
work done in the expansion.

Doing Work on the System and Changing
the System Energy from a Molecular
2.4 Level Perspective
Our discussion so far has involved changes in energy for macroscopic systems, but
what happens at the molecular level if energy is added to the system? In shifting to a
molecular perspective, we move from a classical to a quantum mechanical description
of matter. For this discussion, we need two results that will be discussed elsewhere in
this textbook. First, as will be discussed in Chapter 15, in general the energy levels of
quantum mechanical systems are discrete and molecules can only possess amounts of
energy that correspond to these values. By contrast, in classical mechanics the energy is
a continuous variable. Second, we use a result from statistical mechanics that the
relative probability of a molecule being in a state corresponding to the allowed energy
values e 1 and e 2 at temperature T is given by e -([e 2 -e 1 ]>k B T) . This result will be
discussed in Chapter 30.
To keep the mathematics simple, consider a very basic model system: a gas con-
sisting of a single He atom confined in a one-dimensional container with a length of
10. nm. Quantum mechanics tells us that the translational energy of the He atom
confined in this box can only have the discrete values shown in Figure 2.5a. We
lower the temperature of this one-dimensional He gas to 0.20 K. The calculated
probability of the atom being in a given energy level is shown in Figure 2.5a. If we
now do work on the system by compressing the box to half its original length at con-
stant temperature, the energy levels will change to those shown in Figure 2.5b. Note