Job's Method for Determination of Stoichiometry

Job's Method, also called the Method of Continuous Variation, is a simple and effective approach to the determination of chemical reaction stoichiometry. We will discuss it in the context of generic reaction (1),

which can be rewritten in the form of (2) by dividing all coefficients by "a".

where k = b/a and m = d/a. Job's method is based on the following fact: if a series of solutions is prepared, each containing the same total number of moles of A and B, but a different ratio, R, of moles B to moles A, the maximum amount of product, D, is obtained in the solution in which R = k (the stoichiometric ratio). To implement Job's Method experimentally, one prepares a series of solutions containing a fixed total number of moles of A and B, but in which the R is systematically varied from large to small, and measures the amount of product obtained in each solution. One then plots amount of product versus R, and obtains a maximum at the initially-unknown value of k.

That the maximum amount of product should occur at the stoichiometric ratio can be justified both intuitively and mathematically. The intuitive justification runs as follows. When R is greater than k, there is an excess of reagent B, so reagent A is the limiting reagent. As R is systematically decreased toward k (i.e., as moles A increases and moles B decreases such that moles A + moles B stays constant) the amount of product increases with the amount of limiting reagent, A, until R becomes equal to k. In contrast, when R is less than k, there is an excess of reagent A, and B is limiting. As R is systematically increased toward k (i.e., as moles B increases and moles A decreases), the amount of product increases with the amount of limiting reagent B, until R becomes = k. Putting this all together, we see that as R is varied over the range from zero to the maximum value investigated, the amount of product obtained increases until R = k, then decreases as R becomes larger than k. This demonstrates Job's Method intuitively.

The mathematical justification is also quite simple. We use variable "x" to represent the moles of A in a particular solution, and assume that the total moles of A and B is to be kept at 1.0 throughout the series of solutions. Then in each solution it will be true that

Our goal is to show that the maximum amount of product is obtained when R = moles B/moles A = (1-x)/x is equal to k. We approach this by finding the value of x that maximizes product.

According to equation (2), if x is less than the stoichiometrically correct amount of A, then A is limiting and moles product = mx. A plot of moles product versus x over a series of solutions should be linear, with slope m. Similarly, if x exceeds the stoichiometrically correct amount of A, then B is limiting and moles product = m(1-x)/k. A plot of moles product versus x over a series of solutions should also be linear, with slope = -m/k. The first plot will proceed up to the right as x increases. The second plot will proceed down to the right as x increases. At some point then, the two straight lines will intersect. At the intersection, they have a point in common. The value of x corresponding to this point is obtained by equating the ordinate values and solving for x:

Because the amount of product increases as k is approached from either direction, the point of intersection of the lines occurs at the maximum amount of product obtainable. We have therefore shown that maximum product is obtained when R = k. This is what we set out to demonstrate.

Example: A and B are known to react to form D, but the stoichiometry is uncertain. A Job's Method study yields the following data. Plot quantity of product versus moles A to determine the stoichiometry.

The plot is shown in Figure 1. The value of k is clearly 4, because at the maximum, moles B/moles A = 1.6/0.4 = 4.