A solution is made by adding 4.0 grams of NaOH, which has a molar mass of 40 grams per mole, to enough water to make 1.0 liter of solution. What is the approximate pH of the resulting solution?
We’ve been tasked with finding the pH of a solution, which is defined as the negative log of the concentration of hydronium, or H₃O⁺ ions. When we make our solution, as described in the problem, the NaOH will dissolve and form Na ions and OH ions. Because the sodium hydroxide will form a OH⁻, or hydroxide ions, in water, that makes sodium hydroxide a base.
If we remember our pH scale, a pH of seven is a neutral solution, and anything above that is basic, while anything below that is acidic. Since we know that our solution of sodium hydroxide is a base, it will have a pH that’s greater than seven. So, we know answer choices A, B, and C are incorrect without even having to do any calculations. But we still want to calculate the pH of our solution.
And to do that, we’ll need the concentration of hydronium, which is not something that we’re given in the problem. To find the concentration of hydronium, we’ll make use of the fact that in all aqueous solutions, water can react with itself in a process known as autoionization, or self-ionization. In this equilibrium reaction, water reacts with itself to form hydronium ions and hydroxide ions.
The equilibrium expression for this reaction is the concentration of hydronium times the concentration of hydroxide. The equilibrium constant for this reaction, 𝐾 𝑤, is equal to one times 10 to the minus 14 at 25 degrees Celsius. We can use this expression to solve for the concentration of hydronium, and then we’ll be able to get the pH.
Since the 𝐾 𝑤 is a constant, the only thing we need before we can solve for the concentration of hydronium is the concentration of hydroxide. So, let’s first solve for the concentration of our sodium hydroxide solution, which the problem said was made with four grams of sodium hydroxide in one liter of solution. Just a reminder, that concentration is given in units of molarity, which is defined as moles per liter.
We made our solution with four grams of NaOH. And to find the concentration, we’re going to need to convert this to moles, which we can do by dividing by the molar mass of NaOH. This puts us in moles of NaOH. So, to find the concentration of NaOH, we just need to divide by the one liter of solution. This gives us a concentration of 0.1 molar.
If you wanted to, you could do this calculation in two steps. In the first step, you would get the amount of NaOH in moles by dividing the mass by the molar mass. This would’ve given you 0.1 moles of NaOH, which you would then divide by one liter to get the concentration. Either way, you get the same answer.
Now, we want to find the concentration of the hydroxide ion in our solution. Since according to our balanced chemical equation, there is one mole of sodium hydroxide for every one mole of hydroxide ions, the concentration that we calculated is also the concentration of the hydroxide ion in our solution. Now, we can solve for the concentration of hydronium. We can create an expression for the concentration of hydronium from our equilibrium expression from the autoionization of water by dividing both sides by the concentration of hydroxide.
The 𝐾 𝑤 for water is one times 10 to the minus 14. And the concentration of the hydroxide ion that we just calculated is 0.1. If you want to make the math a little bit easier to do in your head, you can convert the concentration of hydroxide into scientific notation, which is one times 10 to the minus one. If you have two numbers that are exponents that have the same base that you want to divide, you can simply subtract the numbers in the exponents.
In this case, we have a 10 to the minus 14 and a 10 to the minus one. So, the concentration of hydronium is one times 10 to the minus 13 cause [minus] 14 minus a minus one is minus 13. Now, we can find the pH of our solution by taking the negative log of the concentration that we just calculated. To make this math easier to do in our heads, let’s take a minute to think about how logs work.
Logs, unless specified otherwise, are base 10. So, they effectively undo exponents that are 10 to the power of something. So, this log would cancel out the power of 10, leaving behind what was in the exponent, which is a negative 13. And a negative times negative 13 is just 13, which matches answer choice D. So, our NaOH solution has a pH of 13.