Question

The greatest increase in the reaction rate for the reaction between \(\mathrm{A}\) and \(\mathrm{C},\) where rate \(=k[\mathrm{~A}]^{1 / 2}[\mathrm{C}],\) is caused by (a) doubling [A] (b) halving [C] (c) halving [A] (d) doubling [A] and [C]

Short answer

a) doubling [A] b) halving [C] c) halving [A] d) doubling [A] and [C] Answer: d) doubling [A] and [C]

Step-by-step solution

Understand the rate equation

The rate equation \(rate = k[\mathrm{A}]^{1/2}[\mathrm{C}]\) tells us that the rate of the reaction between A and C depends on the concentrations of A and C, with the concentration of A being raised to a power of 0.5. The constant k is the rate constant, which does not change with the concentrations of A and C.

Consider each option individually

We will now consider each option given, and calculate the change in rate as a result of the changes in concentration: (a) doubling [A]: The new rate will be \(k[2\mathrm{A}]^{1/2}[\mathrm{C}]\). Since the square root of 2 is approximately 1.41, this will result in a \(1.41\) times increase in the rate. (b) halving [C]: The new rate will be \(k[\mathrm{A}]^{1/2}[(1/2)\mathrm{C}]\). The rate is halved in this case. (c) halving [A]: The new rate will be \(k[(1/2)\mathrm{A}]^{1/2}[\mathrm{C}]\). Taking the square root of 1/2 is approximately equal to 0.71. The rate will decrease to approximately 71% of its original value. (d) doubling [A] and [C]: The new rate will be \(k[2\mathrm{A}]^{1/2}[2\mathrm{C}]\). As explained in option (a), doubling [A] alone results in a \(1.41\) times increase in rate. Additionally, doubling [C] results in a \(2\) times increase. Therefore, the new rate will be increased by \(1.41*2 = 2.82\) times.

Analyze and determine the greatest increase

By comparing the changes in rate for each option: (a) 1.41 times increase (b) 0.5 times decrease (c) 0.71 times decrease (d) 2.82 times increase We can conclude that the greatest increase in the reaction rate occurs when we double the concentrations of both A and C (option d), with a total rate increase of approximately 2.82 times.