This problem is providing a reaction whereby nitrogen dioxide is produced from nitrogen monoxide and oxygen. Given the rate law, we are asked to quantify the increase in the rate when the concentration of oxygen is tripled. At the end, the result turns out to be an increase by a factor of 3 as well.
In chemistry, chemical kinetics is a tool for us to quantify increases or decreases (among others) in the rate of a chemical reaction, when the concentration or even temperature are manipulated. In such a way, with this reaction's rate law:
[tex]rate=k[O_2][NO]^2[/tex]
One can see it is first-order with respect to oxygen and second-order with respect to nitrogen monoxide. Now, since the concentration of oxygen is tripled, we can setup an r2 to r1 ratio, being the former the reaction after the increase in the oxygen's concentration and the former the initial one:
[tex]\frac{r2}{r1} =\frac{k_2[O_2]_2[NO]^2_2}{k_1[O_2]_1[NO]^2_1}[/tex]
Thus, since the both of the rate constants are the same as no temperature change is pointed out, and the concentration of nitrogen monoxide is not said to have changed, we can cancel them out:
[tex]\frac{r2}{r1} =\frac{[O_2]_2}{[O_2]_1}[/tex]
Hence, we plug in the given increase (tripling):
[tex]\frac{r2}{r1} =\frac{3*[O_2]_1}{[O_2]_1}\\\\\frac{r2}{r1} =3[/tex]
Therefore the increase is done by a factor of 3, which makes sense because the reaction is first-order with respect to oxygen.
Learn more about rate laws: https://brainly.com/question/13309369
