Question

In lecture today, my instructor informed us that...

Answer 1

If you take the log of a number (most specifically the concentration of H+ in a solution), any number to the left of the decimal place is insignificant. For instance:
\[pH=-\log_{10}(3.42*10^{-10})=-9.466\]For this value of pH, there are only 3 significant figures present, and he described that the "-9" is really an exponent.

However, he explained that the "-9" is insignificant. I don't quite understand this. Perhaps my understanding of "significant digit" is incorrect.

For example, if you know the following number to a significance of 3 digits...\[3.645892\]You would report the value as 3.65. All of the "insignificant" digits could be any value, and therefore we are not justified in reporting them. Transferring this knowledge to what my instructor told us, the ones place of a pH value calculated from the [H+] could be any value.

Can anyone elaborate on this?

Answer 2

he's using different words than I would, but the integer portion of pH is really indicated by the order of magnitude. Since the order of magnitude is 10^(-10), the pH will be around 10 anyway.

Answer 3

Yep. I think I understand that portion. But I'm primarily unsure of how a number FURTHER left than other numbers can be insignificant. If the 9 is insignificant, why wouldn't the entire value be insignificant?

Answer 4

I'd have to guess that integer portion of the pH is really more of a "placeholder" than anything else, so the only digits that matter are the decimals, but I could be wrong. I'd ask your prof to clarify.

Answer 5

Only digits to the RIGHT of the last significant digit can be "any value." So, yes, if you report a pH of 3, you are giving an answer with "zero" significant digits, in the sense that the leading digit in [H+] could be anything -- any value between 1 and 9, e.g. any number between 0.001 and 0.009. What you do NOT mean is that digits to the LEFT of the thousandths place could be anything at all!

Be careful to distinguish between the everyday meaning of "significant" and the scientific meaning of "significant" in this case. That is what may lead to the confusion. You're thinking that "zero significant digits" has some profound meaning, like the number could be anything between -infinity and +infinity. But the number of sig digs just is a shorthand way of specifying the precision in powers of 10. 1 sig dig means your precision is 1 part in 10^1, or +/- 10%, and 2 sig dig means your precision is 1 part in 10^2, or +/-1%, and so 0 sig dig just means your precision is 1 part in 10^0, or +/- 100%.

In theory it's even possible to imagine a measurement that has "negative" significant digits, e.g. a precision of -2 sig dig would mean an answer precise to +/- 100x, meaning you could be off by as much as two orders of magnitude. In short, the scientific meaning of "significant" in this context is a lot less significant than you may think, ha ha.