What would be the amount (in g) of Span 80 in 5 g 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) in the mixture. The total mass of the surfactant system is 5 g.

2. Determine the amount of Tween 80 The amount of Tween 80 will therefore be ( 5 - x ) grams.

3. Identify the HLB values

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

4. Set up the HLB equation The total HLB of the mixture can be calculated using the following equation:

$$ \text{Total HLB} = \frac{(HLB_{Span} \cdot x) + (HLB_{Tween} \cdot (5 - x))}{(x + (5 - x))} $$

5. Substitute the values into the HLB equation To achieve a total HLB of 13, plug in the known values:

$$ 13 = \frac{(4.3 \cdot x) + (15 \cdot (5 - x))}{5} $$

6. Multiply both sides by 5 This removes the fraction:

$$ 65 = (4.3 \cdot x) + (15 \cdot (5 - x)) $$

7. Distribute and simplify Now simplify the equation:

$$ 65 = 4.3x + 75 - 15x $$

Combine the ( x ) terms:

$$ 65 = 75 - 10.7x $$

8. Solve for ( x ) Rearrange the equation:

$$ 10.7x = 75 - 65 $$

Simplify:

$$ 10.7x = 10 $$

Now, divide by 10.7:

$$ x = \frac{10}{10.7} \approx 0.93458 , \text{g} $$

9. Round to two decimal places The result is rounded for practical purposes:

$$ x \approx 0.93 , \text{g} $$