MARCH, 1948
a
EXPERIMENTS SIMULATING FIRST AND SECOND ORDER GAS REACTIONS C. C. COFFIN Dalhonsie University, Halifax, Nova Scotia
IT
IS DIFFICULT to h d examples of homogeneous gas reactions suitable for experimental study in undergraduate physical chemistry. The necessary apparatus is in general complex and fragile and requires continual attention. The experiments demand close supervision and are apt to be so time-consuming as to make it impossible for a student to accumulate enough data to illustrate fully the relationships involved. Reactions in solution, also, are likely to require so much analytical work that the kinetic picture remains obscure and incomplete. This paper describes a simple apparatus with which a student can rapidly obtain pressure-time data exactly similar to those afforded by clean-cut and reproducible first and second order association reactions in the gaseous state, and which are worked up in exactly the same way.
FIRST ORDER REACTIONS-FLOW ORIFICE
THROUGH AN
If tv, be the time required for the pressure to fall to one-half its original value (i. e., the half-life of the process), then: 2.3 log2
=
k&/,
and tx/, = 0.693/k,
These equations are characteristic of a reaction in which a gas a t partial pressure P is disappearing according to a first order process. The graph of log Po/Pvs. t is a straight line, the slope of which, multiplied by 2.303, is equal to k,, a velocity constant (frequency) analogous to the specific rate of a fist order chemical reaction. The numerical value of k, is determined by the dimensions of the apparatus and the molecular weight of the gas, so that the molecular weights of diierent gases may be compared directly through their rate constants measured in the same apparatus, i. e.,
In this apparatus a &st order reaction is simulated M I km' by the flow of gas (air, COz, etc.) through a small hole Mf;=!iF from an.otherwise closed volume into a continuously evacuated space. Rate of pressure change a t the hole SECOND ORDER REACTIONS FLOW THROUGH A simulates reaction velocity. If P represents the pres- CAPILLARY sure, t the time, and & the root mean square velocity of The fundamental eauation relating to the viscous the molecules, then: flow of gases through capillary tubesfs that of Poiseuille, where k, and k' are proportionality constants, the values of which are determined by the dimensions of the apparatus and the molecular weight of the gas, M. Collecting variables and integrating, -In P = k,t + I where I is the constant of integration. When t = 0, P =.Po,and I = -In Po,so that:
where V is the volume passing through in t seconds, P is the effective driving pressure, r is the radius, and L the length of the capillary, and q is the viscosity coefficient of the gas. Since V = nRT/P,where n is the number of moles, the above equation may be written:
Maxwell's equation for the viscosity of an ideal gas is: P k,t and log -2= P 2.303
q = l/apzil
where p is the density, 1 the mean free path, and
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the velocity of the molecules. Since p varies directly the effusion orifice A and the capillary tube B, and and 1 varies inversely as the pressure, it is evident from which serves as the short arm of thc one-way manometer this equation. that q is independent of the pressure. The differential equation that should hold for the flow of gas through a capillary tube to a continuously evacuated volume is therefore:
where the constant kt has the value (m4)/(8LRTv), and P is the pressure of 'the gas, i. e., the pressure difference between the two ends of the capillary. Integrating this equation, and evaluating the integration constant by putting P = Powhen 1 = 0 gives:
and
These are the equations characteristic of a second order gas reaction of the type 2A -+ C (or A B -+ C where A and B have the same initial pressure) in which Cis a nonvolatile product. If (PO - P)/PoP is plotted against time, the result is a straight line, the slope of which is the "velocity constant," or "specific reaction Seconds rate," with the dimensions pressure-' seconds-'. Fie.. 2. Eff".l.m of A% and Carbon Dioxide. e;Higher Po: 0.L o r - P With the same capillary a t the same temperature, the values of ka for different gases are inversely proportional to their viscosities which may thus be compared in this MI. The orifice is a small hole punched with a needle in a piece of thin platinum foil cemented over the end apparatus. of a glasst ube; the capillary is 6 cm. of broken thermomEXPERIMENTAL eter tubing cemented into the pyrex apparatus as A diagram of the apparatus is given in Figure 1. shown. For convenience in assembling, the flask is The volume VI is a 250 cc. conical flask which contains connected with the rest of the system by a ground glass joint a t the level AB. In carrying out an experiment the whole apparatus as far as the tap C is evacuated, and the tap H is closed with the mechanical oil pump left running. The gas or vapor is now admitted through C from storage D, or from the volatile liquid in the bulb E, to the desired pressure, which is indicated by the manometer Ma. After closing C, the tap F is turned so that all its three tubes are closed, the tap G is turned to close its "east" and connect its "north" and "west" tubes and H is opened to the pump. As soon as the connecting tubing is evacuated (about 1 mm.), and the mercury ceases to rise in MI (which now indicates the pressure in VI), F is turned to connect VI with the running pump through either the orifice A or the capillary B as desired. The mercury in MI now starts to fall, and, after a few seconds to allow the manometer bscillations to die out, the stop watch is started as the mercury passes the reading chosen as PO. Further manometer. readmgs are made at convenient t , b e intervals throughout the run, which is usually followed for several "half-lives." Since only one side of manometer M I is read during the experiment, these readings must be converted into true uressures before anv calculations are made. A correction curve is reachy oh.tained by plotting readings of MI against those of Ma
+
~
~
MARCH. 1948
when V I and Vz are connected through G and H and the left-hand side of MI is evacuated. V2is a 2-liter volume added to increase the capacity of the low pressure side of the system. The zero of the scale of M I is adjusted to coincide with the position of the mercury in the narrow arm when the pressure is the same on both sides.
0.04
0.08
RESULTS
First Order Reactions. Table 1 gives the results of two typical effusion runs a t different initial pressures for each of the gases-air, oxygen, and carbon dioxide. The initial pressures, first order "rate constants" and "half-lives" are listed in columns two, three, and four: respectively. Column five gives the molecular weights relative to air, calculated from the rate constants (or half-lives).
Run Air 1 Air 2 0%1 Oa 2 CU I co, 2
TABLE 1 Typical Effusion Experiments PQ(m.Hg) kl (see.?) 111,(see.) 1.16 X lo-' 600 '
66.59' 34.33)
:i go.oz
0.01
M (28.95)
1.10 X 10-2
632
32.1
0.95 X
733
43.2
0
,,",3.
Seconds Viscous flow ofAir and Csrbon Dioxide.
e.H i p h e Po;0. ." O L
higher, and the points 0 the lower, initial pressure. The process is clearly second order. As already pointed out these second order rate conIn Figure 2 log Po/Pis plotted against time for the stants for d i e r e n t gases should be inversely proporair and carbon dioxide runs. In each case the points tional to their viscosities. The ratio in Table 2 for the Q represent the higher, and the points 0 the lower, kl values of CO2:air is 3.272.55, which is 1.28; mean initial pressure. It is evident that the process is the generally accepted viscosities a t room temperature strictly first order, and that the time to half (or any are in the inverse ratio 1.23. other) value is independent of the initial pressure. The form of effusion apparatus described above does Second Order Reactions. In Table 2 are listed the not give good values for the molecular weights of subresults of five typical viscosity runs with each of the stances having vapor pressures at room temperature of gases, air and carbon dioxide. Columns two, three, and less than about one-third of an atmosphere. This is four, give, respectively, the initial pressures, second probably because the pump does not keep a sufficiently order velocity constants, and times to half value. low or steady pressure on its side of the orifice, and thus Column five gives the product of initial pressure and introduces an error that becomes larger as Pobecomes half-life and brings out clearly the fact that the two are smaller. In a modification that is much more satisfacinversely proportional to one another. tory for molecular weight determinations, the vapor effuses from a constant pressure (vapor pressure of the TABLE 2 liquid) into a volume where the rate of pressure rise is Typical Viscosity Experiments measured. This is essentially a method that has been Run POicm. Ho) 1x1. (sec.1 P. X 111. described in detail by Eyring.' The usual types of reaction rate problems (e. g., calculation of the fraction disappearing in a specified time interval, prediction of the time required for a certain pressure change, etc.) may be easily built around these flow processes. The distinction between the order and the molecularity of reaction is also well illustrated. Thus viscous flow can in no sense be thought of as a bimolecular process, although it is clearly second order. Effusive flow, on the other hand, is not only a first order process, but may be regarded as unimolecular, in the sense that the moleFigure 3 shows the quantity (Po - P)/PoP plotted cules independently arrive a t and pass through the against time for the viscosity runs, air 3 and 5 and car- hole without interference from their neiehbors. bon dioxide 2 and 3. Again the points @ represent the 1 EYEING, H., J . Am. Chem. Soc., 50,2398 (1928).
Po.
