# Is it possible to find the ratio of isotopes only given the mean mass number?

Three isotopes of an element have mass numbers $(M)$, $(M+1)$ and $(M+2)$. If the mean mass number is $(M+0.5)$, then the ratio of the amounts of the three isotopes is?

Let amount of $(M)$, $(M+1)$, and $(M+2)$ be $x$, $y$ and $z$ respectively. $$\frac{(M)x + (M + 1)y + (M + 2)z}{x + y + z} = (M + 0.5)$$

After further simplification, I got $x-y=3z$.

I don't know how to proceed further. Can someone give me a hint, or reassure me that this question can't be solved with the given data?

The answer is given as $x:y:z=4:1:1$.

You have two equations and 3 unknowns, so you can't solve it with just that. Say a, b, c are the fractions (as a decimal) of each isotope...

$$a(x) + b(x+1) + c(x+2) = (x+\frac{1}{2})$$

$$a + b + c = 1$$

The 4:1:1 solution works. Another that works is 3:0:1. Another is 7:4:1. There are infinitely many solutions.

As already stated, the system of equations is underdetermined. But we can get the range of possible solutions. Starting with $$xM + y(M+1) + z(M+2) = M + \frac{1}{2}$$ and using the normalization constraint $$x + y + z = 1$$ we get $$(x+y+z)M + y + 2z = M + \frac{1}{2} \\ y + 2z = \frac{1}{2}$$ which can be rearranged to $$z = \frac{1}{4} - \frac{y}{2} \\ y = \frac{1}{2} - 2z$$ We can plug $z$ back into the normalization constraint and solve for $y$ $$x + y + \frac{1}{4} - \frac{y}{2} = 1 \\ y = \frac{3}{2} - 2x$$ which in turns allows us to express $z$ in terms of $x$ $$z = \frac{1}{4} -\frac{1}{2}(\frac{3}{2} - 2x) \\ z = x - \frac{1}{2}$$

We further know that $x$, $y$ and $z$ are between 0 and 1. Therefore we can derive: $$0 \le y \le 1 \Rightarrow 0 \le \frac{3}{2} -2x \le 1 \\ \frac{3}{4} \ge x \ge \frac{1}{4}$$ and similar from $z$ we get $$\frac{1}{2} \le x \le \frac{3}{2}$$ Combining all constraints we get all possible solutions for $x$: $$\frac{1}{2}\le x \le \frac{3}{4}$$

• Maybe strict inequalities are more appropriate, since 1:1:0 and 3:0:1 don't really satisfy the claim that the element has the three isotopes. Jul 10 '17 at 1:14
• It is more general this way. It's just a minor detail, and everybody can decide on its own, depending on how the interpret the question: How do we know it's 3 isotopes? Is it just an assumption or did someone actually do a qualitative analysis of the sample? It's not stated in the question .... And as long as you only "claim" there are 3 isotopes, somebody can still prove you wrong ;) Jul 10 '17 at 8:09