What would be the amount (in g) of Span 80 in 5g of a surfactant system composed of a mixture of Span 80 (HLB=4.3) and Tween 80 (HLB=15) to produce a total HLB value of 13?

Steps to Solve

1. Define the variables Let ( x ) be the amount of Span 80 in grams and ( 5 - x ) be the amount of Tween 80 in grams, since the total amount of surfactants must equal 5 g.

2. Set up the HLB equation The overall HLB can be calculated using the formula: $$ \text{Total HLB} = \frac{(x)(\text{HLB of Span 80}) + (5 - x)(\text{HLB of Tween 80})}{5} $$

   Substitute the HLB values:

• HLB of Span 80 = 4.3
• HLB of Tween 80 = 15

3. Insert the values into the equation Substituting the values into the HLB equation: $$ 13 = \frac{(x)(4.3) + (5 - x)(15)}{5} $$

4. Multiply through by 5 to eliminate the fraction Multiply both sides by 5: $$ 65 = (x)(4.3) + (5 - x)(15) $$

5. Distribute the terms Expand the equation: $$ 65 = 4.3x + 75 - 15x $$

6. Combine like terms Rearranging gives: $$ 65 = 75 - 10.7x $$ So: $$ 10.7x = 75 - 65 $$

7. Solve for x Calculate: $$ 10.7x = 10 $$ $$ x = \frac{10}{10.7} \approx 0.9346 $$

8. Round to suitable precision The calculated amount of Span 80: $$ x \approx 0.93 \text{ g} $$