A chemist mixes a 10% saline solution with a 20% saline solution to make 500 milliliters of a 16% saline solution. How many milliliters of each solution does the chemist mix together?

2 Answers
Jun 13, 2017

200" "mL of 10% solution.
300" "mL of 20% solution.

Explanation:

To solve for two unknowns we need two equations/pieces of information. Our two unknowns are the volumes of each stock solution. Let:

x="volume of " 10%" "(mL)
y="volume of " 20%" "(mL)

For our first equation, we know the total volume is 500 mL and is the sum of x and y:

x+y=500" "mL

rArry=500-x" "(1)

For our second equation, we do a mass balance for 500 mL of final solution.

16%" w/v"=0.16" "g/(mL)

This means that in 1 mL of solution, we have 0.16 g of NaCl.

For any solution, concentration multiplied by volume will give the mass of NaCl:

"mass in " x" " mL=C*V" "(g/(cancel(mL)))*cancel(mL)

So in 500" mL" we have 0.16*500" " (g/(cancel(mL)))*cancel(mL)=80" "g of NaCl.

So, the sum of the masses of NaCl in x and y must equal 80" "g, leading to the following expression:

C_x*V_x+C_y*V_y=80
0.1x+0.2y=80" "(2)

Now, substitute our expression for x, (1), into (2):

0.1x+0.2(500-x)=80

0.1x-0.2x+100=80rArr0.1x=20rArrx=200" "mL

Now solve for y using (1):

y=500-x=500-200=300" "mL

Jun 13, 2017

A different approach! Very detailed explanation given.

For the 20% constituent: 3/5xx500= 300" millilitres"
For the 10% constituent: 2/5xx500=200" millilitres"

Explanation:

color(blue)("Preamble about method")

The total volume is a fixed amount. So if the proportion of the 20% concentration is known then the amount of 10% solution is:

" "color(brown)("volume of 10% = total fixed volume - volume of 20%")

Thus by just focusing on the 20% the amount of the 10% is indirectly linked. Thus in this approach we can (sort of) forget about the amount of 10% mix.

By varying the amount of the 20% mix the saline content of the whole changes. It is this change that we are looking at.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
color(blue)("Determine the proportions of each constituent")

Tony BTony B

Let the proportion of the 20% solution be x

The gradient of part is the same as the gradient of the whole.

Using ratio:" " ("proportion of 20% concentration")/("saline content in blend") ->1/(20-10)

Giving

1/(20-10)-=x/(16-10)" " where -= means proportional to

1/10=x/6

Multiply both sides by 6

6/10=x

x=3/5 of the whole
......................................................................................
Thus there is:

3/5 of the 20% material

2/5 of the 10% material

Check:

[3/5xx20%] + [2/5xx10%] = 12+4 = 16%
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color(blue)("Determine the volume of each constituent")

For the 20% constituent: 3/5xx500= 300" millilitres"
For the 10% constituent: 2/5xx500=200" millilitres"