Two-mixture balance
C₁Q₁ + C₂Q₂ = Cᶠ(Q₁ + Q₂)
The amount of pure substance is conserved during ordinary mixing.
Scenario comparison
Store multiple outcomes locally and compare them.
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Formula reference
Core equations used by this solver.
Dilution
C₁V₁ = C₂V₂
Adding pure solvent changes volume and concentration, but not solute.
Repeated replacement
Cₙ = Cᵣ + (C₀ − Cᵣ)(1 − R/V)ⁿ
Each cycle retains the same fraction of the previous mixture.
Evaporation
Cᶠ = (C₀Q₀) / (Q₀ − E)
This assumes the solute is nonvolatile and the solvent evaporates.
Average price
Pᶠ = (P₁Q₁ + P₂Q₂) / (Q₁ + Q₂)
The target price is a quantity-weighted average.
Multi-ingredient blend
Σxᵢ = Q and Σ(cᵢxᵢ) = CQ
Bounds and fixed amounts restrict feasible ingredient combinations.
How to use this calculator
1. Choose a solver mode
Use two-mixture for ordinary concentration equations. Choose dilution, replacement, evaporation, cost, alloy, or multi-ingredient mode when those assumptions fit better.
2. Select the unknown
Several modes provide a solve-for menu. Leave the selected unknown blank when convenient, then enter every required known value.
3. Confirm units
All quantities within one equation must use the same compatible unit. Concentration values should follow the chosen concentration unit.
4. Review validation
Impossible targets, negative quantities, invalid percentages, and incompatible bounds produce clear messages before a result is accepted.
5. Inspect the working
Read the formula, substitution, calculated result, and verification steps. Use the chart to understand ingredient proportions.
6. Export or save
Download CSV data, print to PDF, copy results, save scenarios in local storage, or compare multiple calculations.
Worked examples
Understanding mixture problems
Mixture problems are conservation problems. A component entering a process must either remain, leave, react, or transform. This calculator focuses on cases where the selected component is conserved. The weighted amount of that component is therefore the central quantity.
Concentration may describe mass fraction, volume fraction, mole-based concentration, parts per million, or another ratio. The ordinary percentage modes treat concentration as a fraction of the total mixture. Molarity and molality require additional chemical assumptions, so they are displayed as entered unless a dedicated chemistry conversion is supplied.
Weighted averages
A final concentration is not usually the simple average of two percentages. It is weighted by quantity. Ten liters of a strong solution influence the result more than one liter of that same solution. The calculator multiplies each concentration by its amount, adds those contributions, and divides by the total amount.
Dilution versus mixing
Dilution is a special mixture problem where the added material contains no target solute. The pure substance amount stays fixed while total quantity increases. Mixing combines two or more streams that may all contain the target component.
Evaporation assumptions
The evaporation mode assumes only solvent is removed. It is suitable for many simplified saltwater, syrup, and concentration exercises. It should not be used when the target substance is volatile, decomposes, crystallizes, or reacts during heating.
Replacement processes
Withdrawal and replacement problems are geometric. Every withdrawal removes the current concentration, not the original concentration. Repeating the operation multiplies the retained fraction many times. A nonzero replacement concentration is supported through the recurrence formula.
Bounds and optimization
The multi-ingredient tool applies minimums, maximums, and fixed amounts. It then adjusts a feasible blend toward the target concentration. The lower-cost objective prefers cheaper ingredients when multiple adjustments are available. For regulated industrial formulation, verify the output with a dedicated linear-programming package and process specifications.
Frequently asked questions
Can the target concentration lie outside both source concentrations?
Not in an ordinary two-mixture problem with nonnegative quantities. The final weighted average must lie between the lowest and highest source concentrations.
Why did the calculator return a negative amount?
A negative mathematical amount usually means the target is impossible using the selected ingredients. The application converts that condition into a validation message.
Can I mix mass units and volume units?
Only when density data is available. This single-file solver requires one common quantity basis for each calculation.
Does the dilution mode account for volume contraction?
No. It uses the standard additive-volume classroom model. Real mixtures can contract or expand, especially alcohol-water systems.
How are repeated replacement cycles calculated?
The retained fraction after one cycle is one minus removed amount divided by total amount. That fraction is raised to the cycle count.
Can the multi-ingredient solver minimize cost exactly?
It uses a practical bounded adjustment method. It prefers cheaper transfers but is not a full general-purpose simplex solver.
How do CSV and PDF exports work?
CSV is generated by the PHP file. PDF output uses the browser print dialog, where you can choose Save as PDF.
Where are saved scenarios stored?
Saved scenarios and comparisons use browser local storage. They remain on the current device until cleared.