Relative Abundance and Isotopic Mass?

Question

How do you find the relative abundance and the isotopic mass of an element? Zirconium has four naturally occuring isotopes.
90 = amu 89.905 and % 51.45
91 = amu 90.906 and % 11.22
92 = amu 91.905 and % 17.15
94 = amu ?? and % ??

I need to know the relative abundance and the isotopic mass and how to reach the answer.

I know that the equation is set up like this:
91.224 = 89.905(.5145) + 90.906(.1122) + 91.905(.17.15) + x(1-.5145 + .1122 + .1715).
I just don't understand how to do the math. I keep coming up with different answers every time. I haven't had algebra for two years so I'm not sure what to do with the x.

Answer 1

The easiest thing to do is work with what you know to get a simpler equation.

first, multiply out the terms that don't have x associated with them and simplify what is inside the bracket.

91.224= 46.25612+10.19965+15.76171
+x(.2018)

Next, simplify the non- x terms into one.

91.224=72.21748+x(.2018)

Finally subtract 72.21748 from 91.224 and divide that result by .2018

19.00652= x(.2018)

x=94.185 which is your mass.

your abundance is the fraction we just divided by converted into a percentage.

.2018 = 20.18%

Answer 2

You have a bucket of 100 marbles. The mass of all the marbles together is 1140 grams, which means the average mass of a marble is 11.40 grams.

There are four different types of marbles.

Red marbles weigh 10 grams
Yellow marbles weigh 11 grams
Green marbles weigh 12 grams
You don't know how much blue marbles weigh.

If the bucket contains 50% red, 10% yellow and 15% green marbles, what is the percentage of blue marbles and how much do they weigh?

The percentages add up to 100, so 100 - (50 + 10 + 15) = 25. That means 25% of the marbles are blue.

If the total weight of the bucket is 1140 grams, that total is made up of:
50 red marbles weighing 10 grams each (50 * 10)
10 yellow marbles weighing 11 grams each (10 * 11)
15 green marbles weighing 12 grams each (15 * 12)
25 blue marbles of unknown weight 'x' (25 * x)

The sum of the individual weights must equal the total weight:

(50*10) + (10*11) + (15*12) + (25*x) = 1140

Solve for x. Take it one step at a time.

500 + 110 + 180 + 25x = 1140
790 + 25x = 1140
25x = 1140 - 790
25x = 350
x = (350/25) = 14

and you're done. Blue marbles weigh 14 grams.

Now do the same thing with the weights and relative proportions of zirconium atoms.