Two chemicals, A and B, are combined, forming chemical C. The rate of the reaction is jointly proportional to the amounts of A and B not yet converted to C. Initially, there are 200 grams of A and 300 grams of B, and, during the reaction, for each gram of A used up in the conversion, there are three grams of B used up. An experiments shows that 75 grams of C are produced in the first ten minutes. After a long period of time, how much of A and of B remains, and how much of C has been produced?
A) 200 grams of A, 0 grams of B, 300 grams of C
B) 0 grams of A, 0 grams of B, 500 grams of C
C) 100 grams of A, 0 grams of B, 400 grams of C
D) 0 grams of A, 100 grams of B, 400 grams of C
E) 0 grams of A, 200 grams of B, 300 grams of C

Question

Quizplus · Accepted Answer

Let x be the amount of A converted to C, and 3x be the amount of B converted to C.
Thus, the total amount of C produced is x + 3x = 4x.

The rate of the reaction is jointly proportional to the amounts of A and B not yet converted to C.
Thus, we can write:

(rate of reaction) = k(A)(B)

where k is some constant of proportionality.

Since the rate of reaction is given by the amount of C produced per unit time, and the initial amounts of A and B are 200 and 300 grams respectively, we can write:

(rate of reaction) = (amount of C produced) / (time) = 75 / 10 = 7.5

Substituting all the given values into the equation above, we get:

7.5 = k(200)(300)

Solving for k, we get:

k = 0.000125

We can now use this value of k to find the amount of A and B that remains after a long period of time.

Let y be the amount of A that remains after all the C has been produced, and 3y be the amount of B that remains.
Thus, the total amount of A and B that remains is y + 3y = 4y.

Since the rate of reaction is jointly proportional to the amounts of A and B not yet converted to C, we can write:

(rate of reaction) = k(y)(3y) = 3ky^2

At the end of the reaction, all the A and B would have been converted to C. Thus:

200 - y = x
300 - 3y = 3x

Substituting x and y in terms of C, we get:

x = 200 - y
3x = 300 - 3y
4x = 500 - 4y
4x = 4(125) = 500

Thus, the total amount of C produced is 500 grams.

Substituting this value of x in terms of y, we get:

200 - y = x = (500 / 4) = 125

Solving for y, we get:

y = 75

Thus, the amount of A that remains after a long period of time is 200 - y = 200 - 75 = 125 grams, and the amount of B that remains is 3y = 3(75) = 225 grams. The amount of C produced is 500 grams. Therefore, the answer is (C).

Two Chemicals, a and B, Are Combined, Forming Chemical C