[Lisa_JO]

Data in the table below are for the hydrolysis of an ester. What
must be the concentration of ester at 30 min?

time, min    [ester], M
0                 0.840
15               0.420
30               x
45               0.210


My attempt: The reaction is a 2nd order since there's a linear relationship between t and 1/[A]
Therefore, I used the equation t1/2=1/k[A]0 solved for K with the t/12=30 and  [A]0=0.840
then I used the integrated rate law equation for 2nd order rxn and solved for [A]t which gave me a value of 0.42 but the answer is 0.280

[Corribus]

Well you should know right off the bat that your answer has to be wrong, since 0.42 was the concentration at 15 min (as given), so it can't be the same value 15 minutes later.
Since you have identified the trend as linear (good job), the appropriate way to do the analysis would be to solve for k using a linear regression, then use k and the given value of A₀ to calculate A at t = 30 min. Your lack of subscripts here and lack of writing out what you used as a the rate law makes it difficult to follow your calculation and pinpoint your error, though.

If you don't have access to computer software to do a linear regression, you could simply do a 2 point analysis to approximate k. Using the 15 min time point I got a value of k ~ 0.075 M^-1 min^-1. Then putting that back into the rate law with t = 30 min gives me a value of A ~0.29 M at 30 min. The difference between this answer and your stated answer of 0.28 M is probably due to rounding and the fact that k was determined from only 2 points rather than 3.

[mjc123]

Therefore, I used the equation t1/2=1/k[A]0 solved for K with the t/12=30 and  [A]0=0.840

But t1/2 is not 30, is it? It's 15, as you can see from your data.
You may say: "But the concentration drops from 0.42 to 0.21 over 30 minutes." Yes, but [A]0 for that period is 0.42, not 0.84.
The point is that for a 2nd order reaction, unlike a 1st order one, the half life depends on the initial concentration, and therefore changes during the reaction. The second half-life is twice as long as the first, and the third twice as long as the second, and so on. The third half life will be 60 min, and the concentration will be 0.105 M after 105 min.
That's why you need to be careful about using half-lives to calculate k for non-first-order reactions. Better to use the slope of the 1/[A] vs t curve. This is also likely to be better if it involves more than 2 data points.