DERIVATION OF THE EQUATION P V
=
RT*
LEON McCULLOCH Westinghouse Research Laboratories, East Pittsburgh, Pennsylvania
"THE Equation of State of a Perfect Gas," by Roseman and Katzoff,' recalls a method of derivation of the equation PV = RT which came to the writer after reading the following question from Professor Bancroft : "Why should not the student be shown how to cornbine 8 = KT and P = IT so as to get Pv = RT? The most obvious way of combining them gives PV = RT2 which we know is not right."% w e may COmnence with the two statements from experiments with gases : Y = KT (9 constant) p = kT (v constant)
which are to be combined into one equation connecting the three variables p, v, and T . We may write them thus:
-
* Scientific Paper NO.763. 1 R. ROSEMAN AND S. KATZOPF, "The equation
of state of a perfect gas,'' J. CHEM.EDUC.. 11, 3504 (1934). W. B ~ ~ c n o abook r , review in J . phys. them., 33, 1118 (1929).
.,
v = F(p)T =f(v)T
P
in which the constants K and k have been replaced by functions of p and v. Each equation expresses now a relationship between p, o, and T,these relationships being dependent upon the form chosen for F@) and f(v). We must h d forms for these functions such that the two equations may become simultaneously true. We take T as a variable dependent for its value upon 9 and v, which we shall treat as independent variables. We then consider the two equations as simultaneously true and combine them by eliminating T: P
V
%T=
= f-(4
vf(4 = PFW
We now reason from this equation as follows: The left-hand side of the equation contains only one variable, v , and the right-hand side only the variable quantity 9. These two variables are independent of each other, consequently each side of the equation va-
ries independently, if at all. But the two sides of an equation-cannot vary independently, therefore, there can be no variation; that is, they are constant. We write them equal to the conventional constant R:
w
Or,
=
RT
In a similar manner, we could have proceeded from the two statements: " = - X Y
From these we find the required form of the two functions : 8
F(P)
R
=-
P
Substituting, we obtain Y
=
R P
F(P)T = - T
P
= KT
considering 9 and T as the independent variables, and eliminating v. The method of derivation suggested here is purely algebraic, and is perhaps as nearly as possible self-contained. The reasoning may be difficultfor some, since the conceptions of variables which are independent, of equations which are simultaneously true, of functions of a variable, and of functions which remain constant, are not easily and immediately formed. Thanks are due Messrs. E. B. Ashcraft and R. C. Mason for helpful criticism and suggestions.
CORRESPONDENCE SOME STUDENT EXPERIMENTS IN THE HETEROCYCLIC SERIES-CORRECTION
PRIMARY AMINES FOR USE IN ORGANIC LABORATORY COURSES To the Editor N. BEVERLEY TUCKER Virginia Military Institute, Lexington, Virginia
MANY elementary manuals of organic chemistry, in taking up the study of primary amines, use a dilute solution of methylamine prepared from acetamide by Hoffman's hypohalite method. I t is suggested that the experiments usually performed with this solution might well be supplemented by some with a liquid amine which can be handled in concentrated solution. For this purpose, ethylenediamine is suggested. I t can be purchased fairly reasonably in sixty per cent. aqueous solution. It shows quite satisfactorily the usual reactions of a prima~yamine, such as the formation of an odoriferous isocyanide in solution, precipitation and re-solution of silver and cupric hydroxides, precipitation of femc hydroxide, etc.' I t reacts violently with acid sodium nitrite. The hydrochloride and sulfate are easily prepared by simply adding the concentrated acid to the diamine solution diluted with alcohol. The formation of a substituted amide by heating the ammonium salt may be illustrated by adding glacial acetic acid to the diamine solution and distilling to a temperature of about 175'C. The residue of diacetyl ethylenediamine2 is best purified by recrystallization from a mixture of ethyl acetate and alcohol. Many other simple experiments suggest themselves. FISCRER a m Kocn, "Reactions of diamines," Ann., 232,222
,--",. 119%)
a HOPPW, "Notiz iiber Anhydrobasen der aliphatischen Diarnhe," Ber.. 21, 2332 (1888).
DEARSIR: A letter from Drs. C. B. Pollard and L. G. MacDowel1 draws attention to an error in the paper, "Some Student Experiments in the Heterocyclic Series," [J. CHEM.EDUC.,13, 265-72 (1936)], by the undersigned. On page 271 the name "N-Pbenylpiperidine" appears instead of "N-Phenylpiperazine." The writers regret this slip and hasten to correct it. The communication from Messrs. Pollard and MacDowell states further that a "practical" grade of diethanolamine, available at a moderate price, is. satisfactory for the preparation of N-phenylpiperaqine, and that the procedure outlined in their original note [J. Am. C h m . Soc., 56, 2199 (1934)], when amplified in certain details, has given good results in the hands. of their students.
A PLAN FOR THE OPEN HOUSE IN
CHEMISTRY-CORRECTION To the Editor DEARSIR: In the article "A Plan for the Open House in Chemistry," [J. CHEM.EDUC.,13, 72-3 (1936)], the table showing the approximate composition of a one-hundredpound girl should have included the item: slaked lime, 4.5 pounds. The content of calcium fluoride should be changed to 25 g.
