MATHEMATICAL PROBLEM PAGE Directed by EDWARD L. HAENISCH Montana State College, Bozeman, Montana
I
T IS gratifying to know that this page has a SUB- Since one atmosphere = 76 a n . of mercury, cient number of interested readers to warrant its revival. The present director hopes that he can maintain the high standard set by Dr. Daniels and Dr. Cross. = 1,013,200 (gm.cm. see.-') em.-' = dynes It is planned to present each month a hrief "lesson" cm.? followed by a selection of exercises of varying diiculty. and The solutions will be published the month following. Suggestions for various topics for the "lessons" will be 1,013,200 dynes an.-a X 22,400 cm.8 R = welcome. 1 mol X 273 deg. * * * = 8.315 X 10' dyne an. mo1.-I deg.-I = erg Many experiments in chemistry involve the measuremol-I deg.-l. ment of some quantity or other. The result must include not only the name of the quantity measured and Since 4.185 X lo7erg = 1 cal. its magnitude but also the units in which it has been measured. A systematic study of units and their uses R = 1.987 cal. mol-I deg.-1. is well repaid. The most practical system of units is that adopted by I n t e r n a t k l Critical Tables, for it recognizes the unit of A (re4 temperature and the chemical unit, the mol, as well as C(onductance) = S(pecificConductivity) Uength) the fundamental units of mass, length, and-time. I t will be remembered that the "MLT" dimensions comS = CL/A = ohm-' an.-'. monly assigned do not include these first two quantities. The units of a quantity are most easily obtained from The choice of the proper units is of the utmost imthe defining equation. The simplest physical quauti- portance. One would obviously not use R in cal. mol-1 ties are listed in Table 1. deg.3 in the equation, P V .- nRT, when P is measured in atm. and volume in The choice becomes more TAB== 1 dif6cult in an equation of the type: Qmnrilr Length *. m a .. . ~ Volume Velodty Acceleration Momentum Force Preewre work Density Molec. Weight Speei6e Heat Heat Capaelty
Definilion
u n i o in icrmr ofan., gm.. rrc.. tnal., nnd dcn.
Name of "nil
em.'
DistaneeITime
VelotitylTime Mass X Velocity Mass X Acceleration ForcdArea Force X Distance Mars/Volurne
Fm.l
em. * e : . cm. set-2 gm. gm. see.-1 gm. cm. see-: gm. om.-,
gm. rnolFL
-.-'
cm.2 see.-' gm. em': gm. mol-1 eal. deg.'! gm.-1 Ern.
dyne dynes cm ': erg
cal. deg.? mol-1
Pressure is commonly stated in atmospheres (atrn.), as well as in dynes an.-=. To the electrical quantities we will assign the units in which they are usually measured, namely: current, ampere; potential, volt; resistance, ohm; quantity, coulomb. The units of several more complicated quantities will now be determined: (a)
=
nRT
1 atm. X 22,400 R = mol X 273 dex.
To make the units of the right side of the equation agree with those of the left R must be in gm. cm.' see.-' mol-' deg.-'. Closer examination shows this to be erg mol-I deg.-I. As will be seen from the above, an equation can be tested for its correctness by demonstrating the agreement or disagreement of the units on both sides of the equation. As an illustration, suppose that
were falsely remembered as the approximate form of the Clausius-Clapeyron equation. Checking the units,
The Gas Constant.
PV
u = root mean square velocity of a molecule
em.
=
atm. cal. mol-I deg.' - deg.8 -= - deg. cal. mol-' deg.? atm. atm.
82.06 an.%tm. mol-1 deg.-I.
shows its inaccuracy. 42
Here V denotes the volume of liquid of v i s c ~ s i t ~ , ~ ~ , flowing through a capillaty tube of length, 1, and radius, r, in the time, t, and under the pressure. P.
Finally, unit analysis can be relied upon to tell the form of variance of one quantity with another. It is known that the velocity of a compression wave (sound wave) in a gas varies as the pressure and density of the gas; i. e., Y = f(p,d). Suppose w varies directly as the pressure and inversely with density. We have, if p is measured in dynes
(e)
Here t, is the &tical temperature of the liquid whose surface tension is 7, a t the temperature of the experiment, 1. M is themolecular weight and u is the specific volume.
p- ~ (gm. cm. LC.-=) - dynes ~ m . -d
=
gm. ~ m . - ~
gm. an.-' cm.= set.?
Thisshows us that u = d g d , for cm. ~ec.-'adcm.~sec.-~.
(1) Macg I
The constant of the Ramsay and Shields' modification of the EBtvBs equation:
~ FRANCE, D "Laboratory
manual of physical chemistry," D. Van Nostrand Co., Inc., New York City, 1934, pp. 1-12.
2.
Calculate a value of K, the gas consrant, if pressure is m a s ured in pounds per square foot and vulwnc in cubic feet.
3. Are the following equations correct except for their constant factors?
(2) International CritiGal Tables, Vol. I , pp. 1C32.
where w is a weight, a is an area, t is time, and R, T have their usual significance.
p. M.
PROBLEMS
1. What are the units of the following quantities? (a) a in Van der W d s ' equation (h) Equivalent conductivity (c) d(bg 9) if 9 is measured in atmospheres (d) Vismsity. (Use Poiseuille's equation for the flow of liquids through capillary tubes.)
,,=
where v is the drift speed of an ion, Lois the limiting equivalent conductivity of that ion, E is the potential gradient, Fis Faraday's constant. 4.
WP r' -
d in Kcc varies with dT
A H , R, and T.
Determine the form of
the variance by dimensional reasoning.
8V1
.. AN IMPROVED HEATED VACUUM MICRO-DESICCATOR EUGENE W. BLANK Col~ate-Palmolive-PeetCo., Jersey City, New Jersey
*
in operation. The improved desiccator is shown in Figure 1. The desiccator tube has been fitted with a alass ioint I at (A) to simplify the operation of chargin;! and reI l + b moving the de&ant. It is advisable to &an interchangeable joint if possible so that the desiccator may be kept in continuous service if need arises. The thermometer has been placed within the desiccator instead of in a well bored in the aluminum block. - 1 This insures a more accurate temperature control, but the principal advantage secured is that the upper half of the aluminum block may be removed more readily, .m. c and without disturbii~the thermometer, to observe the condition of the ma&ial within the desiccator. T o I ~ Gl.-Cnosss~cr~o~ ~ E OR iw IMPROVED HEATED VACUUM end it is to have a ring attached to the xt.nma-n-e7m,.aTa* upper half of the aluminum block to faditate handling. In the interim several modifications have been made Other details of the desiccator remain essentiallv the in the original design tending toward greater advantages same. For a fuller description of the apparatus the reader is referred to the original paper already cited. 1 BLANK,J. -M. EDUC.,10, 189 (1933).
A VACUUM micro-desiccator heated by means of a micro-burner was recently described in THISJOURNAL.'
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