A NONRADIOACTIVE DEMONSTRATION OF THE ISOTOPIC EXPONENTIAL EXCHANGE LAW
0
MILTON KAHN University of New Mexico, Albuquerque, New Mexico
Tm equation
rotation of stopcock D will result in a net transfer of air from B to A and that eventually the pressures in AX BXo = AXo BX both sections will be equal. represents a simple isotopic exchange reaction where If a given number, n, of 180° turns of stopcock D the atom X is common to molecules AX and B X ; Xo represents an interval of time t, then nu represents I., indicates a radioactive atom of X . Neglecting radio- the rate of transfer of "volume" from B to A and from active decay of Xo, the exponential exchange law for A to B. Provided v is very small compared to the this reaction is' volumes of A and B, the rate a t which the pressure in A will increase with time is given by the following ex- ~n (1 - F) = a+b ~t ab (2) pression:
+
+
where
-1" (I - P A )-V
F = fraction exchange; a = concentration of X (active inactive) in the form of AX (moles.liter-!); b = oncentratmn of X (active inactive) in the form of BX (moles .liter-'); t = time; R = constant rate of exchange of X a t o w between AX and BX (moles .liter-'. time-').
+
FRIEDLANDER, G., AND .J. W. KENNEDY, "Introduction to Radiochemistry," John Wiley & Sons, Ino., New York, 1949, p. 285.
+ V a rl
VA
(3)
where V~ = vO1umeof
+
In this paper an experiment is described whereby a student in the undergraduate physical-chemistry laboratory can demonstrate the operation of the exchange law without the use of radioactive isotopes. This experiment has the particular value of forcing the student to correlate observations on a system apparently far removed from isotopic exchange, with the ideas presented in the derivation and discussion of the exchange law as it pertains to radioactive exchange. Consider the apparatus shown in Figure 1. The straight bore of stopcock D has been plugged so as t o leave only a small volume v, in the bore a t each end. Thus, by turning stopcock D through 180' a volume u of gas in section B is transferred to section A and a similar volume of gas is transferred from A to B. Initially, by proper manipulation of stopcocks F and C, the entire system is filled with air at some arbitrary pressure. Section B is isolated from A by turning stopcock F; A is then evacuated and subsequently isolated from the pump by manipulation of stopcock C. It is clear that continued
=
A; V B = volume of section B; P A = pressure in section A at time t; P'n = Pressurein at time = 0.
Equation (3), which is similar in form to equation (2), is derived below in a manner designed to show the analogy between the previously discussed experiment and isotopic exchange. The derivation is parallel with that given by Friedlander and Kennedy for the isotopic exchange law! Imagine each section ( A and B ) to he composed of a large number of cells of volume AV. Each cell can accommodate one and only one molecule of gas. Cells differ only in that they may or may not contain a molecule of gas. In the chemical system the radioactive isotope is treated as chemically identical with the stable isotope. I n the gas system a cell containing a molecule is equivalent to an empty cell with respect to transfer from one section to another. Moreover, T O MANOMETER
TO PUMP
r
1.
~ w - ~ ~ APP-~W h a ~ .
C,three-why stopeook; D. 6-mm. straight-bore stopcock; E, plug:
in
F . 2-mm. straight-bore stopcook.
JOURNAL OF CHEMICAL EDUCATION
178
I = fraction of cells in A which are "marked" and also the probability that a cell transferred from A to B is "marked"; -- ! I = fraction of cells in B which are not "marked" and also the probability that a cell transferred from B to A is not "marked": = probability that in a given exchange of cells a a b "marked" cell in A is exchanged far a cell in B which b not "marked"; -Rz'b - !I = rate of transfer of "marked" cells from A to B a t a b any given instant; = rate of appearance of "marked" cells in A.
dt
I t follows that
The solution of this first-order linear equation is:
where C is a constant of integration. When t
u 2,
1
2
3 Time
4
a
5
-
Unit of time is 50 360' turns. Upper curve: V A = 226 eo., I'D = 177 Lower curve: Y1 = 77.2 cc.. VB 177 co.
oc.
a cell which contains a molecule is a "marked" cell and corresponds t o the radioactive isotope in the chemical system. The following symbols and expressions of Friedlander and Kennedy are redefined in terms of the gas system: = total number of cells in A; b = total number of cells in B; number of "marked" oells in A; y = number of "marked" cells in B; z =z y = total number of "marked" cells. R = number of cells exchanged between A and B in unit time;
Z
or in words the ratio of ''marked" a f b cells in A to total number of cells in A equals the ratio of the "marked" cells in A and B to the total number of cells in A and B. Therefore, the second term on the right of equation (5) equals x,, the number of "marked" cells in A when the pressure in A equals the pressure in B. Also, when t = 0, x = 0 because initially A was evacuated, and consequently C = -x,. From the foregoing discussion it follows that: m - = -
0
=
j
' a
a
Equation (6) is identical with equation (2) where
1- =
F
+
#
= fraction of cells in B which are "marked" and also the probability that a cell transferred from B to A is "marked"; a- z = fraction of cells in A which are not "marked" and a
also the probability that a cell transferred from A to B is not "marked"; = probability that in a given exchange of cells a b a "marked" cell in B is exchanged for a cell in A which is not "marked"; 3 = rate of transfer of "marked" cells from B to A at b a any given instant;
Results of Gas-exchange Experiments
Po#, mm.
V A ,FC.
Unit of time equals 50 360" turns.
V B ,CC.
T,cC./P
-.x
Equation (3) is obtained from equation (6) x, by considering that: a
=
=
v n / a v ,b
=
-
V B / A V ,R = r/AV, z PA, and 2, Y V B P 0 ~ / V " V B
+
EXPERIMENTAL
The 6-mm. bore of stopcock D was plugged with a small, tighkfitting solid rubber cylinder. Apiezon wax Q was applied to both sides of the rubber plug with a hot glass rod in order to insure against leakage. In practice it is difficult to plug the stopcock so that the volumes at each end of the bore are the same; t h e r e fore, a rotation of D through 360' was considered as effecting an exchange of volume equal to the total volume in the bore. Stopcock, D was lubricated with Apiezon grease L. It was found desirable to replace the grease after about 700 360" rotations because the stopcock became increasingly difficult t o turn. The manometer was constmcted of 2-rnm.bore capillary tubing in order to minimize the slight change in the volume of A as the pressure in A increased. The volume of B was calculated from the weight of water re-
APRIL. 1955 quired completely to fill B. The volume of A was then determined by simply allowing air at a known pressure in B to expand into A which had been previously evacuated, and noting the final pressure of air in the entire system. Some experimental results are summarized in the table. Typical plots of in (1 - F) versus tare shown in Figure 2. These plots are straight lines leading into the origin in accordance with equation (3). The results show that the rate of exchange r is independent of PB,the initial pressure in B. Analo-
119
gously, in the chemical system the rate of exchange
R is independent of the amount of radioactivity present. The average value of r, 18.0cc./t, obtained via the exchange experiments is in good agreement with the value 17.5 cc./t calculated from the total volume of the bore which was obtained by determining the weight of mercury required completely to fill the bore. It should be noted that, whereas in the gas system r is not a function of V A or VB,R in the chemical system will, in general, depend on a and b.
