edited by
J. DUDLEY HERRON Purdue Untverstty West Lalayette. Indiana 47907
The following outline for a lesson on gases arrived with a request for help with a teaching problem. We have provided some possible solutions to the teacher's quandry; but we would slso like to hear from our readers. Interesting suggestions will be used in a future column.
The Problem: "Derivation of The Ideal Gas Law" John J. Vaitkunas Stuttgart American High School APO 09154' New York, N.Y. Let me begin by quickly developingthe approach that I take with my students. We develop the gas laws via a "functional" approach: fh).After describing.some of the physical .. . i.e....v = . . . properties of gases-volume, pressure, temperature, mass, kinetic energy and so on-we ask several questions concerning these properties. I) The volume of a gas is influenced by what physical properties? Another way of saying this is: Volume is a function of what physical properties? V = f ( ) Students generally acknowledge that volume is influenced by pressure, temperature and number of particles (mass). Therefore: V = f(P,T,n)
(1)
The same eeneral auestion is asked about pressure,. tempernture and so on. 2) The Dressure of a eas - is influenced by what .physical . properties? P = f(V,T.n)
(2)
3 ) The temperature of a gas is influenced by what physical properties? T = f(P,V,n)
Solving for the k constants: PPV=k*
T - k ~
If volume is in liters, pressure in tom,temperature in kelvin, and n in moles, the specific units of the k constants may be written. k,=PV=torr.l
k 2 = P- = -torr T K
h a = P- -- - torr n mole
The three relationships mentioned ahove are used in developing the "ideal gas law."
However, this is where the difficulty arises. How does one go from these three separate relationships to the ideal gas law? Several college texts that were consulted did not provide a detailed derivation. Somehow the following expression seemed to be ohtained. nT P = k,k2k3-
v
(7)
Assuming k l X k2 X ks = R p = -RnT
(8)
v
I t seems that the equation should he $,3
RnT =-
v
(9)
This is proven if k , X k z X ks is evaluated from the individual relationships described ahove. In unit form,
However, the conventional units of R are torr 1/K mole. Could you provide an explanation or refer me to some work or text that would explain the derivation?
Some Suggested Approaches
The error in logic in the ideas presented in this lesson plan occurs in going from eqs. ( 6 )to eqn. ( I ) ,because the only way that the product klk2h3 can appear in eqn. 7 is to multiply eqns. ( 6 ) ,which gives nT P3 = k,k2k3-
v
530 1 Journal o f Chemical Education
(5)
(3)
After identifying dependent variables, independent varinhles, and variables that need to be held constant, rough craphs are drawn to show mathematical relationships. This is especially useful, for it helps the student to visualize the relationship between factors. I t also allows them to see why certain variables must be held constant when developing a relationship between the dependent variable and the independent variable. The variables held constant are noted at the top of the graphs in the figure. The mathematical relationship is expressed below the graphs in the figure. These are made into equalities with the introduction of a :onstant. Then the constant is solved for in terms of its dimensions or units.
jimpie graphs showing relationships involving pressure.
P
- = k3 n
(10)
This, of course, is equivalent to eqn. (9). There are three obvious routes for providing assistance in this lesson plan. 1) If we are to remain with the empirical gas relationships, Charles' law ( V = k 2 T ) should he used in place of the second eqn (6).Thus, eqns. (11)and (13)
shows that k2hs does indeed have the same units as R. 2) A second route to assistance involves eqns. (6)which car he read as proportionalities. A well known theorem in algebra states that any quantity (P) which is jointly proportional tc several quantities ( I N , n, and T i n this case) is also proportional to the product of there quantities (eqn. (18)).
can he combined to give
which can he rearranged to
k kan using eqn. (12) to substitute for V and rearranging gives
'=v
However, since h l = PV, from eqn. ( l l ) , it follows that
~
~
or
(kzk3)nT= P V (17) which is of the form of the ideal gas law if k2k3 = R. A check
Expression (18) can he made into an equation by the usual process of introducing a constant which we call the gas constant (eqn. (19)).
Equation (19) is, of course, equivalent to eqn. (8) in the lesson described above. 3) The third route available is to develop the ideal gas equation from the tenents of the kinetic molecular theory. The detailed arguments in this process are available in any one of a series of standard general chemistry texts. Does anyone have another alternative to the problem pmed in this lesson? Send your suggestions to J. Dudley Herron, Department of Chemistry, Purdue University.
Volume 56, Number 8, August 1979 1 531
