The rate of a certain chemical reaction is directly proportional to the square of the concentration of chemical \(A\)

[Gmat_mission]

The rate of a certain chemical reaction is directly proportional to the square of the concentration of chemical \(A\) present and inversely proportional to the concentration of chemical \(B\) present. If the concentration of chemical \(B\) is increased by \(100\) percent, which of the following is closest to the percent change in the concentration of chemical \(A\) required to keep the reaction rate unchanged?

(A) \(100\%\) decrease
(B) \(50\%\) decrease
(C) \(40\%\) decrease
(D) \(40\%\) increase
(E) \(50\%\) increase

Answer: D

Source: GMAT Prep

[Scott@TargetTestPrep]

The rate of a certain chemical reaction is directly proportional to the square of the concentration of chemical \(A\) present and inversely proportional to the concentration of chemical \(B\) present. If the concentration of chemical \(B\) is increased by \(100\) percent, which of the following is closest to the percent change in the concentration of chemical \(A\) required to keep the reaction rate unchanged?

(A) \(100\%\) decrease
(B) \(50\%\) decrease
(C) \(40\%\) decrease
(D) \(40\%\) increase
(E) \(50\%\) increase

Answer: D

Solution:

We can let n = the rate of a certain chemical reaction, a = the concentration of chemical A, and b = the concentration of chemical B. We are given that the rate of a certain chemical reaction is directly proportional to the square of the concentration of chemical A present and inversely proportional to the concentration of chemical B present, so, for some positive constant k, we have:

n = ka^2/b

When b is increased by 100 percent, b becomes 2b. To keep the reaction rate unchanged, we can let a become c, so we have:

ka^2/b = kc^2/(2b)

2bka^2 = bkc^2

2a^2 = c^2

c = √(2a^2)

c = a√2

Since √2 ≈ 1.4, c ≈ 1.4a or approximately 140% of a, i.e., a 40% increase in the concentration of chemical A.

Answer: D