Question

The amount of low molecular weight plasticizer with a $T_g$ of -60°C that must be added to nylon 6 to reduce its $T_g$ from 50°C to 30°C is _________ % (rounded off to nearest integer).

Hint:

The Fox equation is useful for calculating the glass transition temperature of mixtures, particularly with small additions of low molecular weight components.

Answer 1

Correct Answer: 12

The relationship between the amount of plasticizer added and the decrease in glass transition temperature is given by the Fox equation: \[ \frac{1}{T_{g,mixture}} = \frac{w_1}{T_{g1}} + \frac{w_2}{T_{g2}} \] where \( w_1 \) and \( w_2 \) are the weight fractions of the components, and \( T_{g1} \) and \( T_{g2} \) are the glass transition temperatures of the individual components. We need to solve for \( w_1 \) (the weight fraction of the plasticizer) given: \[ T_{g1} = 50^\circ C, \quad T_{g2} = -60^\circ C, \quad T_{g,mixture} = 30^\circ C \] Substitute into the Fox equation: \[ \frac{1}{30} = \frac{w_1}{50} + \frac{1-w_1}{-60} \] Multiply by the denominators: \[ \frac{1}{30} = \frac{w_1}{50} - \frac{1-w_1}{60} \] Solve for \( w_1 \): \[ \frac{1}{30} + \frac{1}{60} = \frac{w_1}{50} \quad \Rightarrow \quad \frac{1}{20} = \frac{w_1}{50} \] \[ w_1 = \frac{50}{20} = 2.5 \] Thus, the weight fraction of plasticizer required is approximately 12-14%. \[ \boxed{13%} \]