The reaction table in the passage indicates that the reaction rate varies in a 1-to-1 fashion as you vary the each reactant, while holding the other constant.

The rate law is written as .

Compare trials 1 and 2 to see that doubling the ammonium concentration doubles the rate. The reaction is first order for ammonium: .

Compare trials 2 and 3 to see that tripling nitrate concentration triples the rate. The reaction is first order for nitrate: .

The final rate law is .

To find the overall order of a reaction, look at the exponents for the reaction rate law. Since A has an exponent of two and B has an exponent of one, the overall reaction has an order of three. 

For a first order reaction, use the equation:

In this formula,  is the concentration at a given time,   is the initial concentration,  is the rate constant, and is the time elapsed.

If we assume the initial concentration to be , then we can use as the final concentration. Using these values and the given rate constant, we can calculate the time.

We can look at the differences between trial 1 and trial 2 to see that the reaction is first order with respect to [A], since the rate doubles when the reactant concentration doubles. Reactants are considered first order when this 1:1 relationship between concentration and rate is observed.

We can look at trials 2 and 3 to see that [C] has no effect on the reaction, making it a zeroth order reactant and simply equal to 1 in the rate law.

Trials 3 and 4 show that when [B] doubles, the rate quadruples. This means it is second order with respect to [B], increasing the rate by the square of the increase in concentration. Since we already know that reactant C does not affect the rate, it is irrelevant that its concentration is halved.

This gives us Rate = k[A][B]².

Reactant concentration, along with activation energy and temperature, is a major determinant in reaction rate, as demonstrated in Le Chatlier's principle.

This question can be approached two different ways. One way is by looking at the equation, and the other is understanding what the variables in the equation represent.  

1) The first method is understanding the equation. Because we must understand basic log and natural log functions, we know that the natural log of a fraction is a negative number. Additionally, because we know that a reaction is spontaneous when  is negative, we can conclude that a reaction will be spontaneous when Q is less than K_eq. R is a positive constant, and T cannot be negative.

2) A second approach to this question is understanding the variables in the equation. K_eq is the ratio of product to reactant at equilibrium. Q is the reaction quotient, and represents the ratio of product to reactant at a given point in time. If Q is less than K_eq, the reaction is still too far to the left, and is still reacting to reach equilibrium. Because the reaction is trying to reach equilibrium, it will continue to spontaneously react until it reaches that point. 

The reaction with the fastest rate will have the lowest activation energy; the energy difference between the starting compound and the top of the energy barrier that must be overcome to proceed to the product should be the smallest. Of the given choices, the reaction proceeding from C to D has the lowest energy barrier, and hence will have the lowest activation energy and fastest reaction rate.

In contrast, the energy required to move from point A to point B has the largest energy barrier, and can be assumed to be the rate-limiting step.

If you thought all of these factors are used in the Arrhenius equation, consider that the rate constant is independent of the reactant concentrations. The concentration effect is considered in the rate law by multiplying the concentrations by the constant, but the concentrations themselves are independent of the actual Arrhenius calculation.

The calculation shows that . In this equation, is the activation energy and is the temperature.

The amount of energy released or absorbed by a reaction is independent of its rate. Rate is principally determined by the activation energy, while the inherent bond energies of reactants and products dictate thermodynamic changes. One cannot be used to predict the other.

Reaction rates, including the rate-determining step, can be altered with a change in conditions. This includes a change in temperature, a change in pressure, or the addition of a catalyst.