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Estimate the ratio of $\ce{Tb^3+}$ to $\ce{Tb^4+}$ in $\ce{Tb_6O_11}$ ? For my solution I knew that I had to get a sum of 22 from 6 $\ce{Tb^3+}$ and $\ce{Tb^4+}$ ions so I just added different combinations until I found that $4(\ce{Tb^4+})+2(\ce{Tb^3+})$ gave me the desired value to have an overall neutral oxidation state on the molecule, this gives a $2:1$ ratio also. Is this correct way to do this or is there a more formal method?

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    $\begingroup$ your way looks good to me $\endgroup$ – DavePhD Dec 15 '15 at 13:20
  • $\begingroup$ Well, trying different combinations is definitely a lot better than just sit there and stare at it. You could go with a linear equation, but with numbers like these it is hardly any faster. $\endgroup$ – Ivan Neretin Dec 15 '15 at 13:20
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There is certainly a formal way. Let $x$ and $y$ be the number of $\ce{Tb^3+}$ and $\ce{Tb^4+}$ ions respectively. Then:

$$\begin{align} 3x + 4y &= 22 \\ x + y &= 6 \end{align}$$

This is a simple system of two linear equations in two unknowns, and I am sure you know of many ways to solve it. Let's just solve it using matrices for the fun of it:

$$\begin{align} \begin{pmatrix} 3 & 4 \\ 1 & 1 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} &= \begin{pmatrix} 22 \\ 6 \end{pmatrix} \\ \begin{pmatrix} x \\ y \end{pmatrix} &= \frac{1}{3\cdot 1 - 1\cdot 4} \begin{pmatrix} 1 & -4 \\ -1 & 3 \end{pmatrix} \begin{pmatrix} 22 \\ 6 \end{pmatrix} \\ &= - \begin{pmatrix} 1\cdot 22 - 4\cdot 6 \\ -1 \cdot 22 + 3\cdot 6 \end{pmatrix} \\ &= \begin{pmatrix} 2 \\ 4 \end{pmatrix} \end{align}$$

as you found. It seems that trial and error may well be faster!

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