Quick question about pH calculations

[Dr.ATS]

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What is a trick for calculating hydronium ion concentration when given a pH?

For example: Say you have a pH of 2.59. I believe the hydronium ion concentration is 10^-2.59. I just do not quite understand how to covert the 10^-2.59 into a usable molarity.

I am stuck on a few Kaplan questions that gloss over how to do this without a calculator and in a timely fashion.

 

[aldol16]

You just need to be in the right ballpark to get the question right. So you'll need sufficient accuracy to rule out the other answer choices but you don't necessarily need the exact number. Sometimes, you'll just need the right order of magnitude! Here's a quick way to do exponents:

10^-2.59 is between 10^-2 and 10^-3. You should be able to do those in your head. 10^-2 is simply 0.01 and 10^-3 is 0.001. So it's between 0.01 and 0.001. The thing with exponents, because 10^x is an upward sloping graph, the more positive you go, the faster you increase in value. For instance, you would think that 10^2.5 is halfway between 10^2 and 10^3. But it's actually much closer to 10^2 because when you plot y = 10^x, you see that as x increases, y increases faster and faster. In other words, as you go from x = 2 to x = 2.5, that's less of an increase in y than from x = 2.5 to x = 3.

So when you're looking at 10^-3 and 10^-2, you can remember this fact - that 10^-2 is more positive than 10^-3 and therefore 10^-2.5 is not the halfway point and will be closer to 10^-3 than 10^-2. In other words, 10^-2.5 is less than 0.005. So using this, you can narrow it down further: 10^-2.59 is between 0.005 and 0.001. This should be sufficient to answer any question. It's okay if you don't follow this last part - the original 0.01 and 0.001 should also be sufficient to answer the question.