Page 51 - Essentials of physical chemistry

Ideal and Real Gas Behavior 13

We use the total pressure of the two gases in the pressure uncertainty since that is what the
manometer is measuring with an uncertainty of 1 mmHg (according to our subjective eye) and
even though we measure the temperature to the nearest degree (low-precision thermometer) we need
to remember we are using the Kelvin temperature in the calculation. Now all the variables are in the
same units of percent. In this case we have three variables so we can plot them on a (P, V, T ) axis
and display the value calculated for the moles as a vector. The relative uncertainties can be displayed
as þ= deviations about the tip of the calculated vector result.
Then we use the 3D Pythagoras’ theorem to estimate the worst case vector displacement of the tip
of the calculated vector result. The result will be in percentage units as

q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
2
2
(0:1333) þ (2:2222) þ (0:3376) ffi 2:2515%
L% ffi
Finally, we can quote the final answer with an estimated uncertainty using the ‘‘square root of the
sum of the squares of the percent uncertainties in the contributing variables’’ [2] to find
mol N 2 ¼ 0:0178 2:2515% ¼ 0:0178 4:0076 10 4 mol

This method is subjective relative to how precise a given observer can measure each variable but
standard precision glassware should be estimated to be good to about 0.1% and included in the sum of
squares for each item of glassware used so this approximate method can be extended to many
variables. We see that according to the uncertainty analysis the calculated result is only good to the
fourth decimal place as given. Some texts express error analysis=uncertainty in terms of partial
derivatives but it is not easy or clear to assign a partial derivative to a response from a given device
in terms of a calculus formula in some cases. For that reason, this author favors the simple formula in
terms of percent uncertainties, which can be (subjectively) estimated numerically.

NONIDEAL GAS BEHAVIOR
While the ideal gas law works well for pressures up to about 10 atm and higher temperatures above
258C, many common processes (air conditioning, refrigeration) involve higher pressures and lower
temperatures. If the ideal gas law is truly universal we could define the ‘‘compressibility factor’’ as

PV
¼ 1
Z ¼
nRT
and expect that if we plot Z against the pressure we should get a flat line (Figure 1.6). When such
graphs are plotted for real data, there are large deviations, particularly at low temperatures and=or
high pressures.
There are other ways to plot these data to exaggerate the deviations from Z ¼ 1, but on the other
hand we can see that over a fairly large range of temperatures and pressures the ideal gas law is
approximately correct. What are the reasons for the deviations from the ideal? Let us try to patch the
ideal gas law for a more detailed treatment. We start by setting up the basic PV behavior and allow
for corrections.

(P þ ? 1 )(V þ ? 2 ) ¼ nRT

Consider a correction to the pressure, P. If indeed the pressure we measure is due to molecular
impacts with a surface in a manometer or a diaphragm in a pressure gauge, is that the actual pressure
within the gas? We are creeping up on a new concept that models a gas as a collection of small