Problem-Solving Strategies

In those note, I enumerate certain strategies and give advice on approaching the midterm.

Understand Your Objective

You may think, most likely subconsciously, that your objective on an exam should be to solve all problems. Wrong. Your objective should be to get as many point as you can. As soon as you understand this, your approach to solving problems should change drastically.

Instead of getting stuck on any given problem and trying to tackle it until you solve it — you just distribute your time evenly over all problems. Imagine you have 50 minutes and 6 problems (fyi, I have no idea how many problems you will have on an actual exam). I’d leave 10-20% of time for review, in this case 8 minutes, and then distribute the rest evenly: 7 min per each problem ($7\times 6+8=50$). Start working on problem 1 and do as much as you can in 7 min. After 7 minutes, it doesn’t matter whether you solved the problem or not. Move on to problem 2. It is hard to do so, you may feel it’s wrong to leave the problem unsolved, you may think oh, if I just spend a few more minutes, I may solve the problem. But which outcome is better, have 2 fully solved problems and receive 0 marks on the other 4, or have partial solution to all 6 problems? Aesthetically, I could agree the former is better. Point-wise, it’s very likely you’ll have a greater score in the latter outcome.

Translate the Problem from English to Math

Consider the following problem:

In a reaction $2\mathrm{A}\to \mathrm{B}+3\mathrm{C}$, the $A$ is consumed at a rate of $0.01\mathrm{M}{\mathrm{s}}^{-1}$. What is the rate at which $B$ is produced?

Something I observed many of you do when solving problems like this, is write "$0.01\mathrm{M}{\mathrm{s}}^{-1}$" in your notes and then try to argue. But then it’s very easy to confuse what this value represents? You may think it’s the reaction rate, and then say okay, the reaction rate expressed through $B$ is $d[B]/dt$, therefore the answer is $0.01\mathrm{M}{\mathrm{s}}^{-1}$, which would be incorrect.

Imagine if I gave you the problem set up as follows:

In a reaction $2\mathrm{A}\to \mathrm{B}+3\mathrm{C}$, the value of $\frac{d[A]}{dt}$ is equal to $-0.01\mathrm{M}{\mathrm{s}}^{-1}$. Find $\frac{d[B]}{dt}$.

I’m confident all of you can solve the problem in this formulation. As long as you understand the following identity:

$$-\frac{1}{2}\frac{d[A]}{dt}=\frac{d[B]}{dt}=\frac{1}{3}\frac{d[C]}{dt}$$

All you need to do is simply plug in the value for $\frac{d[A]}{dt}$!

Before you start solving any problem, your task is to translate it from English into symbolic representation. Do not allow yourself to write any values without labels. Read the problem, find every piece of data presented in the problem, convert it into symbolic form and write it down in your solutions.

Double Check your Interpretation of the Problem

Consider the following problem:

$A$ is a reactant which is consumed in a process obeying first-order kinetics with the rate constant $k=1.05\cdot 10^{-3}{\mathrm{s}}^{-1}$. Find the time in which the concentration of $A$ decreases to one third of its starting concentration.

Similar problem was in one of the worksheets, and a common approach looked like this:

$$\begin{array}{rl}\frac{1}{3}\ln [A{]}_0& =\ln [A{]}_0-kt\\ \frac{2}{3}\ln [A{]}_0& =kt\\ \frac{2}{3k}\ln [A{]}_0& =t\end{array}\text{\hspace{1em}(3.1)}$$

Then you may try to plug in the values and find that you get a negative answer (because the log of a concentration smaller than 1 is negative). But no matter how long you look at your work, you will not find any issues with it. Because there are no algebraic mistakes. The mistake is in the interpretation of the problem. The correct equation should be:

$$\ln \frac{1}{3}[A{]}_0=\ln [A{]}_0-kt\text{\hspace{1em}(3.2)}$$

But I’d argue that you shouldn’t skip preceding steps. The problem, written in English, states neither (3.1) nor (3.2) directly. What it says is:

• $A$ is consumed in the first-order process. Therefore, we can say that $\ln [A{]}_t=\ln [A{]}_0-kt$.
• We need to find $t$ when A decreases to one-third of its starting concentration. Thus, we seek $[A{]}_t=\frac{1}{3}[A{]}_0$. You can obtain (3.2) by plugging the second condition into the equation from the first condition. If you skip this step and you try to write equation 3.2 directly, you may make a mistake and write 3.1 and it’ll be very difficult to find where you made a mistake.

Do not guess. Formulate complete logical statements.

Consider the following problem:

The Haber process $\mathrm{N}{}_2+3\mathrm{H}{}_2\rightleftharpoons 2\mathrm{NH}{}_3+92.4\mathrm{kJ}\mathrm{mol}\text{-}1$ is an exothermic reaction. What will happen to the equilibrium constant if we increase the temperature?

The correct answer can be expressed with a single word, so it may be tempting to immediately try and say that word: decrease. The problem is, you probably won’t have a lot of confidence in your answer. You might as well say increase. How do you know if you’re right? Can you tell whether you’re right by looking at the answer? No.

What you should do is to formulate complete logical statements like this:

The reaction is exothermic, which means that heat/energy is released during the forward reaction. If we increase the temperature, we essentially add energy/heat to the system. The system responds to our efforts by trying to minimize the influence of our efforts (Le-Chatelier principle). Therefore, the system will try to decrease the amount of energy. It can do so through the reverse reaction (if the forward reaction releases energy the reverse must absorb it). Therefore, the system will prioritize the reverse reaction and so the amount of reactants will increase. Because reactants are in the denominator of the equilibrium constant, the value of the equilibrium constant will decrease.

Is it longer? Yes. But it’s also more foolproof. You can check whether you have the correct answer or not simply by evaluating whether each individual sentence is true and then whether the logical connections (because/therefore) are used correctly. With practice (and time) you may be able to start doing this reasoning in your head or even subconsciously. But in the beginning, you should write it out explicitly. Not only because it’ll increase the chance that you find the correct answer, but also because you can get partial credit for your solution.

Imagine if you make a mistake and interpret “exothermic” as heat is absorbed during the forward reaction. Your final answer will be that the value of $K$ will increase, which will be a wrong answer. And if you write just that answer, you will get 0 points. However, if you write the whole paragraph above, it’ll be easy to see that you have the correct reasoning and you made a minor mistake in interpretation of the terminology. So you will get partial credit.

Do not underestimate psychology

If you look at mistakes like above (confusing whether heat is released or absorbed in exothermic reactions) you may think that it’s impossible for you to make such a mistake. But how many times did you find yourself looking at the graded exam/problem set, realizing you made a simple mistake and you couldn’t even fathom how could you make such a mistake?

You shouldn’t underestimate the psychological factors associated with taking an exam under time constraints and knowing that the score you get will impact your grade (and potentially many other things downstream). You can’t avoid those factors, you can only adapt to them by putting yourself in such conditions more often. Whenever you are given practice questions, put yourself in an exam environment: set the timing, remove all distractions, and solve all problems as if it’s an actual exam. This may look like a fairly banal advice, but in many cases the key to success is in banal and simple steps. There’s no secret formula.

As a good test for how much are you affected by psychological pressure, you can first solve all problems with time constraints, then after time runs out, take a pen of different ink and continue solving/reviewing problems until you have the most confidence in your answers. Then grade only the things you wrote under time constraints and compare it to the score you get if you grade everything you wrote after that. By looking at the difference you’ll see the impact of time-constraints on your performance.