What is the rate equation given this data table?

Question

A chemist does a reaction rate analysis on the following reaction:

2NO _(g) + O_{2 (g)} → 2NO_{2 (g)}

She collects the following data:

Trial Initial [NO] (M) Initial [O₂] (M) Instantaneous Rxn Rate (M/s)

1 0.0250 0.0253 0.0281

2 0.0500 0.0253 0.112

3. 0.0250 0.0506 0.0562

What is the rate equation for this reaction?

Answer 1

Compare trials 1 and 2:

[NO] doubles and [O₂] is constant and the rate increases by by 4 fold. This tells us it is 2nd order in NO.

Compare trials 1 and 3:

[NO] is constant and [O₂] doubles and the rate doubles. This tells us it is 1st order in O₂.

Thus, the rate law can be written as ...

Rate = k[NO]²[O₂]

Answer 2

The forward rate equation for any reaction with two reactants can be written as R = k * [A]^a * [B]^b, where R is the rate, k is an arbitrary constant, [A] is the concentration of A, a is the number of A molecules in the slow step (typically just regarded as the coefficient of A in the balanced equation), [B] is the concentration of B, and b is the number of B molecules in the slow step (typically just regarded as the coefficient of b in the balanced equation). This can be expanded to more reactants by adding in C, D, etcetera.

Your goal for this question is to determine all the powers (a, b, c, d....) and k by using R, [A], [B], [C], [D].....

Start by solving for the powers. To do this, you can notice the relative concentrations given in the table. In fact, you can completely forget about the actual ones until you try to solve for k, if that helps you. For example, the first column has a concentration (let's call it [A]), double that (2[A]), and the same as the first ([A]). The simplest way to find the power a is to find two rows in which everything is the same except for [A] and R. These are the first and second rows. Take note of how the concentration of A changes and how that changes the rate. Here when A is doubled, the rate quadruples. From this, you can set up the equation 2^a = 4. You can solve for a by plugging in whole numbers until you make the equation true (for a simple and well-made problem) or by taking the logarithm (the base being 2 here) of both sides (for a hard or real problem). If you would like a more mathematically rigorous way to do this, you can plug in the relative or actual values of what you know in two sets of conditions and divide the equations by each other to cancel many variables. The power b can be solved in the same way here, but it isn't uncommon to see problems where you need to know some of the other powers before solving for the rest. In which case, just plug in the powers you know to the general equation while solving for the ones you don't.

Once you salve for all the powers, you can solve for k. If the problem is well-made, all you need to do is plug in all the concentrations, powers, and rate for any set of conditions (pick the easiest numbers). The equation should simplify out to something like m=nk, where m and n are arbitrary numbers, and all you need to do to solve is divide both sides by n.

Once you are done, you get the equation you want by plugging the powers and k into the general equation (leave R and the concentrations as variables). Hope this helps!