Domain: Polymer networks

Given a rubber that has \(\bar{\text{Mn}}\)  = 100,000 g/mol before crosslinking. After crosslinking, it is stretched to 2 times its unstretched length at 298 K supporting a stress of \( 0.84~\text{MN/m}^2 \). After aging 10 days in air at 398 K, the stress at 2 times its unstretched length at 298K is only \( 0.48~\text{MN/m}^2 \) . If the degree of crosslinking of the rubber before aging is "X" and the fraction of crosslinks lost on aging are "Y"., then calculate the value of (5X -7Y) using the density of the crosslinked rubber (without filler)  which is  \( 0.98~\text{g/cm}^3 \)  and R = \( 8.37 \times 10^7~\text{erg mol}^{-1}\text{K}^{-1} \)

Grading rubric:
criterion: Correctly identifies the Flory chain-end correction factor as the specific geometric modification to the affine network model.
weight: 10
criterion: Asserts the assumption that polymer density remains constant between the unaged and aged states.
weight: 10
criterion: Correctly rearranges the stress equation to solve for the molecular weight between crosslinks (\(M_c\)).
weight: 10
criterion: Correctly calculates the unaged molecular weight between crosslinks to be within the precise range of 4620 to 4625 g/mol.
weight: 10
criterion: Correctly calculates the aged molecular weight between crosslinks to be within the precise range of 7560 to 7565 g/mol.
weight: 20
criterion: Correctly reports the fraction Y as a dimensionless value of exactly 0.389 within a 0.001 tolerance.
weight: 20
criterion: Correctly evaluates the final expression 5X minus 7Y to a negative value within the range -2.720 to -2.725.
weight: 20

Answer:
-2.72