Exact Analytical Solution:

Given:
- Feed flow rate: F = 150 kg/h
- Feed composition: 25 wt% acetone, 75 wt% water
- Solvent flow rate: S = 100 kg/h pure hexane
- Distribution coefficient: K_D = y/x = 1.6
- Water and hexane are immiscible

Step 1: Calculate component flow rates
Acetone in feed: 0.25 × 150 = 37.5 kg/h
Water in feed: 0.75 × 150 = 112.5 kg/h
Hexane in solvent: 100 kg/h

Step 2: Define variables
Let x = mass fraction acetone in raffinate
Let y = mass fraction acetone in extract
From K_D: y = 1.6x

Step 3: Material balances
Water balance: All water reports to raffinate
(1-x)R = 112.5 ⇒ R = 112.5/(1-x)

Hexane balance: All hexane reports to extract
(1-y)E = 100 ⇒ E = 100/(1-y) = 100/(1-1.6x)

Acetone balance: Input = Output
37.5 = xR + yE = x[112.5/(1-x)] + 1.6x[100/(1-1.6x)]

Step 4: Solve the equation
37.5 = 112.5x/(1-x) + 160x/(1-1.6x)

Multiply both sides by (1-x)(1-1.6x):
37.5(1-x)(1-1.6x) = 112.5x(1-1.6x) + 160x(1-x)

Expand both sides:
Left: 37.5(1 - 2.6x + 1.6x²) = 37.5 - 97.5x + 60x²
Right: 112.5x - 180x² + 160x - 160x² = 272.5x - 340x²

Bring all terms to left side:
37.5 - 97.5x + 60x² - 272.5x + 340x² = 0
37.5 - 370x + 400x² = 0

Multiply by 2: 75 - 740x + 800x² = 0
Rearrange: 800x² - 740x + 75 = 0

Step 5: Solve quadratic equation
x = [740 ± √(740² - 4×800×75)]/(2×800)
Discriminant: 547600 - 240000 = 307600
√307600 = 554.616

x = [740 ± 554.616]/1600

Two solutions:
x₁ = (740 - 554.616)/1600 = 185.384/1600 = 0.115865
x₂ = (740 + 554.616)/1600 = 1294.616/1600 = 0.809135 (not physically possible)

Final Answer: x = 0.116 mass fraction acetone in raffinate

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