Question

You mix 335 gallons of a mixture containing 4 wt% NaCl (sodium chloride) and 96 wt% water (solution density = 1.0253 g/cm3) with an unknown volume of solution that is 26 wt% NaCl and 74 wt% water (solution density = 1.1944 g/cm3). You want to produce a mixture containing 14 wt% NaCl. Find the volume of the second feed solution by following the steps below.

Answer 1

239.525 gallons

Step-by-step explanation:

For the mass balance (which comes from Lavoisier rule), the initial mass of a compound must be equal to its final mass less its accumulation in a process. In a mixture, there is no accumulation, so:

(mNaCl)i = (mNaCl)f

Where i is the initial and f the final. For the initial mass, we have two different solutions and we must sum both masses. So, first let's calculate these masses:

mass of NaCl for solution 1:

1 gallon = 3785.41 cm³

335 gallon = 1.268x10⁶ cm³

So, the solution mass is equal to the density multiplied by the volume:

m = 1.0253x1.268x10⁶ = 1.30x10⁶ g

The mass of NaCl corresponds to 4% of the total mass, so:

mNaCl = 5.20x10⁴g

The mass of NaCl in the second solution will be 0.26m, where m is the total mass of the solution. So,

(mNaCl)i = 5.20x10⁴ + 0.26m

The final mass, is them sum of the initial ones, so it is 1.30x10⁶ + m, and the mass of NaCl in this solution must be

(mNaCl)f = 0.14x(1.30x10⁶ + m)

Then,

5.20x10⁴ + 0.26m = 0.14x(1.30x10⁶ + m)

0.26m - 0.14m = 1.82x10⁶ - 5.20x10⁴

0.12m = 1.30x10⁵

m = 1.083x10⁶ g

The volume of the solution is the mass divided by the density, so:

V = 1.083x10⁶/1.1944 = 9.067x10⁵cm³

V = 239.525 gallons