You mix 19.5 gallons of a mixture containing 12 wt% NaCl (sol'n density = 1.0837 g/cm^3) with V2-gallon sol'n that is 20 wt% NaCl (sol'n density = 1.1453 g/cm^3). Needed mixture is 19 wt% NaCl
Whats the V2 in gallons?
(all solutions are in water so just dealing with NaCl)
V1 =19.5gellon=2113.2cm^3
TM1 =2113.2*1.0837= 22900.9 g
M1= 12*TM1= 2748.1 g
V2 =?
TM2 =V2*1.1453
M2 = 20*V2*1.1453
V3=V1+V2 (mixing of two solutions)
Conservation of mass:
M3=M1+M2,
M3= 12*V1*1.0837 + 20*V2*1.1453 ....I
TM3 = TM1+TM2,
M3 = 19*TM3
= 19*(V1*1.0837 + V2*1.1453) ......II
equate equation I and II
(you have value of V1, so its just 1 variable equation)
V2= 129.158 gellons Ans
To solve this problem, we need to use the concept of mass balance. Let's break down the information given:
1. We have 19.5 gallons of a mixture containing 12 wt% NaCl.
2. The density of this mixture is 1.0837 g/cm^3.
3. We are adding V2 gallons of a solution that is 20 wt% NaCl.
4. The density of this solution is 1.1453 g/cm^3.
5. We want to find the value of V2 that will result in a mixture containing 19 wt% NaCl.
Let's start by calculating the masses of NaCl in each part of the mixture:
Mass of NaCl in the initial mixture = volume of mixture x density of mixture x weight percent of NaCl
= (19.5 gallons) x (1.0837 g/cm^3) x (12 wt% / 100) = 25.095 grams
Mass of NaCl in the added solution = V2 gallons x (1.1453 g/cm^3) x (20 wt% / 100) = 22.906 g
The total mass of NaCl in the final mixture will be the sum of the masses from both parts:
Total mass of NaCl = Mass of NaCl in initial mixture + Mass of NaCl in added solution = 25.095 g + 22.906 g = 48.001 g
Now, let's set up the mass balance equation:
Total mass of NaCl in the final mixture = (Total volume of the mixture) x (Weight percent of NaCl in the final mixture)
48.001 g = (19.5 gallons + V2 gallons) x (19 wt% / 100)
Dividing both sides of the equation by (19 wt% / 100), we get:
48.001 g / (19 wt% / 100) = 19.5 gallons + V2 gallons
Now, we can solve for V2:
V2 gallons = (48.001 g / (19 wt% / 100)) - 19.5 gallons
V2 gallons ≈ 10.163 - 19.5 = -9.337 gallons
So, the value for V2 is approximately -9.337 gallons.
Please note that a negative value doesn't make sense in this context, so the problem may have been set up incorrectly or there may be additional information missing.