Question

Question #12ca1

Answer 1

Here's what I got.

Explanation:

The underlying principle of a dilution is that you can decrease the concentration of the solution by

• keeping the number of moles of solute constant
• increasing the total volume of the solution

So, in order to dilute a solution you must increase its volume without changing the amount of solute it contains.

Let's assume that solution `"A"` contains an unknown number of moles of solute, `n_A`. The initial concentration of the solution can be written as

`["A"]_0 = n_A/V_(A0)`

`["A"]_0 = n_A/(0.3 * 10^(-3)"L") = (10/3 * 10^3 * n_A)color(white)(a)"mol L"^(-1)`

After you mix this solution with solution `"B"`, the total volume of the resulting solution will be

`V_"total" = "0.3 mL" + "6.7 mL" = "7.0 mL"`

Keeping in mind the fact tha the resulting solution must contain `n_A` moles of solute `"A"`, you an say that its new concentration is equal to

`["A"]_"dil" = n_A/(7.0 * 10^(-3)"L") = (1/7 * 10^3 * n_A)color(white)(a)"mol L"^(-1)`

So, by what factor, `"D.F"`, was solution `"A"` diluted? Divide the initial concentration by the final concentration to get

`"D.F." = ( 10/3 * color(blue)(cancel(color(black)(10^3 * n_a))) color(red)(cancel(color(black)("mol L"^(-1)))))/(1/7 * color(blue)(cancel(color(black)(10^3 * n_a))) color(red)(cancel(color(black)("mol L"^(-1)))))`

`"D.F." = 10/3 * 7 = color(green)(bar(ul(|color(white)(a/a)23.3color(white)(a/a)|)))`

This tells you that the resulting solution is `23.3` times less concentrated than solution `"A"` compared with the initial concentration of `"A"`.

An interesting thing to notice here is that you can get the dilution factor by dividing the final volume by the initial volume of the solution

`"D.F." = (7.0 color(red)(cancel(color(black)("mL"))))/(0.3color(red)(cancel(color(black)("mL")))) = color(green)(bar(ul(|color(white)(a/a)23.3color(white)(a/a)|))) ->` for solution `"A"`

The exact same approach can be used to find the dilution of `"B"`. This time, the initial volume will be equal to `"6.7 mL"` and the final volume to `"7.0 mL"`

`"D.F." = (7.0color(red)(cancel(color(black)("mL"))))/(6.7color(red)(cancel(color(black)("mL")))) = color(green)(bar(ul(|color(white)(a/a)1.04color(white)(a/a)|))) ->` for solution `"B"`

So, to sum this, the dilution factor for a given dilution is equal to

`color(blue)(|bar(ul(color(white)(a/a)"D.F." = "initial concentration"/"final concentration" = "final volume"/"initial volume"color(white)(a/a)|)))`