How to represent dilution as a formula when the dilution agent is not pure.

Question

So the question we were looking at is as follows:

Start with 70g of Solder that has a 74% content of tin

To this add solder that has a 6.5% concentration of tin

How many grams of the dilution agent (6.5%) must be added. For the overall solution to be at 59% tin content.

I worked through this in a very manual way, and came to the answer 20g

But this was simply achieved by working out the per gram value and then trying 1g, 10g and finally 20g

I have found the dilution formula:

$C_1V_1 = C_2V_2$

But I think this assume the dilution agent is pure.

Any help appreciated.

Answer 1

The simple answer is to create an algebraic equation that tracks the amount of tin. So:

Assume you add $W$ grams of the more dilute solder to the $70$ grams of the more concentrated solder to form the solder with the desired concentration.

1) Can you write an expression for the total weight of solder you'll have?

2) Can you write an expression for the total amount of tin you will have in this final solution? (Remember, it's $56$% tin)

3) Can you write an expression for the amount of tin in the original $70$ grams of solder?

4) Can you write an expression for the amount of tin in the $W$ grams of dilute solder you add?

'Cause $2) + 3)$ must equal $4)$

Edit:

Well, there may indeed be a standard formula, but it's easy enough to create it. Just repeat the process outlined above, but use symbols for all the variables:

1. Dilute solution has percentage $P_{low}$and mass $W_{low}$
2. Concentrated solution has percentage $P_{high}$and mass $W_{high}$
3. Desired solution has percentage $P_{goal}$ and mass $W_{goal}=W_{low}+W_{high}$

Then $$P_{low} \times W_{low}+P_{high} \times W_{high}=(W_{low}+W_{high}) \times P_{goal}$$ Rearranging: $$W_{low}=W_{high} \times \frac{P_{goal}-P_{low}}{P_{high}-P_{goal}}$$

BTW, the solution you found, $20$ g of dilute, yields a final solution of $59$%, not $56$%