Isotopic mass, or maybe just a silly math question

[CleffedUp]

Homework Statement

The atomic weight of bromine is 79.904. The natural abundance of Br-81 is 49.31%. What is the atomic weight of the only other natural isotope of bromine?

Z = 35

Homework Equations

The answer is 78.92 amu (Br-79) and Br-81 happens to be 80.92 amu.

The Attempt at a Solution

Restated, the problem is: 0.4931y + 0.5069x = 79.904

...which amounts to a linear equation, not a single solution for x.

I feel like either I've overlooked some facts/method to derive y (Br-81 amu), or there's some math maneuvering I've forgotten. Before this semester, I was 12 years removed from a classroom -- I have gotten some practice in, but the latter wouldn't surprise me.

Thanks for taking the time.

 

[Matterwave]

y is the mass of Br-81.

Are you supposed to derive both the mass of Br-81 AND the other mass?

It's impossible to do that with the information given. I think the mass of Br-81 should be a given (it should be...81amu).

 

[sjb-2812]

y is the mass of Br-81.

Are you supposed to derive both the mass of Br-81 AND the other mass?

It's impossible to do that with the information given.

Agreed

I think the mass of Br-81 should be a given (it should be...81amu).

Not necessarily, look into binding energy or similar.

 

[epenguin]

Mmm late at night but I'd say that's possible. Even though 'The natural abundance of Br-81 is 49.31%' sounds ambiguous. (49.31 of the mass or 49.31% of the atoms?). Should be defined in textbook but never mind. About half of bromine is of atomic mass 81. The average is close to 80. There is only one other isotope. The atomic masses of isotopes are close to whole numbers. Seems solution obvious? Then do more careful calculation see whether everything fits exactly.

Edit. Sorry that was late at night. I thought you were supposed to get the nearest number atomic mass which is easy. For exact calculation I am not sure what you are give, but from what you write you seem to have been given y.

 

[CleffedUp]

First off, thanks for all the replies.

All that was given is listed under the question, and the textbook answer is the precise mass of the other isotope, so plugging in 81 as y doesn't cut it. If you do the algebra, you get 78.84, which does tell you the other is probably Br-79, but this isn't the precise answer.

... <light bulb> ...

... <TI-36X, engage!> ...

Funny how a passed night and a few restatements of the problem conjures new approaches.

If you do plug in the AMUs into my equation above (Let y = 81, x = 79), you wind up with a number slightly larger than the atomic weight, 79.9862. If you then divide the actual atomic weight (79.904) into that, you get a proportion of 0.99900 (0.998972323... but I'm trying to mind my significant figures). Well don't you know that 0.99900 * 79 = 78.921, which is the textbook answer, and the other isotope checks.

I'll definitely tinker some and take this question to my professor to see if this is some kind of rule, or if Br happens to just work out this way.

All being said, this question is at the end of chapter 2, and there was nothing in the chapter about this sort of approach. However you look at it, I think I just agree that this is just a bad question.

Thanks again!

 

[CleffedUp]

One postscript:

One thing I still don't understand though is why I couldn't come to this answer by multiplying amu by the particle values, e.g.

Br-81 = 35p * 1.0073 + 46n * 1.0087 + 35e * 0.0005 = 81.6732 != 80.92

 

[chemisttree]

Binding energy between protons and neutrons lowers some of the mass.

 

[CleffedUp]

...binding implies "slower," and m = e/c^2, so m is lower! Makes sense. Thanks CT. I'll go off and Google for a deeper understanding.