Coffee has a pH of about 5.1. Calculate to two significant figures the hydroxide ion concentration of coffee at 25 degrees Celsius.
We want to find the hydroxide ion concentration or the concentration of OH−. But we’re given the pH, which is defined as the negative log of the concentration of H3O+. In any aqueous solution, the water in the solution can react with itself in a process known as self-ionization or autoionization. In this equilibrium reaction, the water reacts to form H3O+ or hydronium and OH− or hydroxide.
The equilibrium expression for this reaction is the concentration of hydronium times the concentration of hydroxide divided by the concentration of water squared. Since the concentration of water doesn’t really change significantly, we can remove it from this expression. If we rearrange this equation by dividing both sides by the concentration of hydronium, we’ll arrive at an expression that we can use to solve for the concentration of hydroxide.
The equilibrium constant for this expression, which we call Kw, has a value of 1.0 times 10 to the minus 14 at 25 degrees Celsius. So the only thing we need to find in order to solve for the concentration of hydroxide is the concentration of hydronium, H3O+. The log that we use in the pH is a log base 10. Using log rules, we can undo this log by raising both sides to a power of 10. The first thing that I’ve done is just swapped the sides of the equation so that the log of the concentration of hydronium is on the left instead of the right.
Now, let’s divide by negative one to move the negative over to the other side. Now we can undo the log by raising everything to a power of 10. Now, we’ve obtained an expression for the concentration of hydronium. If we plug the value of the pH in, we’ll find that the concentration of hydronium in coffee is 7.9433 times 10 to the minus six molar.
Now, we can find the concentration of hydroxide. If we plug everything in, we’ll find the concentration of hydroxide is 1.258 times 10 to the minus nine molar. Rounding to two significant figures, the concentration of hydroxide in coffee is 1.3 times 10 to the minus nine molar.