In many industrial and laboratory settings, you may need to adjust the concentration of a liquid mixture to meet specific requirements. This commonly happens with coolants, cleaning solutions, plating baths, or any process where a solute (like a concentrated chemical) is mixed with a solvent (usually water). The basic approach is to calculate how much concentrated solute must be added to an existing solution to achieve a desired final concentration after topping off with solvent.
Understanding the Problem
Consider this example:
A 100-gallon tank is to be filled with coolant at an 11% concentration.
Currently, the tank has 53 gallons of solution at 5% concentration.
How many gallons of pure coolant need to be added before topping off the tank with water?
The Formula
The required amount of concentrate can be calculated using:
Vc = (Vt × Cf) − (V1 × C1)
Where:
- Vc = volume of concentrate to add
- Vt = total final volume of the tank
- Cf = desired final concentration (decimal form)
- V1 = current volume in the tank
- C1 = current concentration (decimal form)
Step-by-Step Solution
Step 1 — Plug in the numbers
Vc = (100 × 0.11) − (53 × 0.05)
Step 2 — Simplify
Vc = 11 − 2.65
Step 3 — Solve
Vc = 8.35 gallons
Answer: Add 8.35 gallons of pure coolant. After adding the coolant, fill the remaining space in the tank with water:
Vw = 100 − 53 − 8.35 = 38.65 gallons
Related Problems
- Starting from Empty
If the tank were empty and you needed an 11% concentration in 100 gallons:
Vc = 100 × 0.11 = 11 gallons of concentrate - Increasing to a Higher Target Concentration
If the tank has 80 gallons at 6% but must reach 10% in a full 100-gallon tank:
Vc = (100 × 0.10) − (80 × 0.06) = 10 − 4.8 = 5.2 gallons - Finding the Final Concentration After Adding Concentrate
If you add 5 gallons of concentrate to 45 gallons of a 5% solution in a 100-gallon tank:
Cf = ((V1 × C1) + Vc) ÷ Vt