SME
Organic Chemistry II · Problem type
Carothers equation: chain length in step growth
Use this for “Nylon-6,6 (113.16 g/mol a residue) is stopped at 98.5% conversion of its end groups. Find Xₙ and Mn.”
Chart
Xₙ(0.99) = 100
This is an example. Type your own values to start a new problem.
Xₙ
=
1
1
−
M₀
Mean residue molar mass
example · g/mol
Mn
Number-average molar mass
calculated · g/mol
Formulas
Xₙ
=
1
1 −
p
100
=
1
1 − 0.99
Mn
=
Xₙ
×
M₀
11,316 = 100 × 113.16
Assumptions
•
The two monomers are mixed in exactly equal amounts, so no end group is left over.
•
Every end group reacts equally fast, whatever the size of the chain it sits on.
•
p is the fraction of end groups that have reacted, and M₀ is the mean molar mass of one residue.
Step-by-step
WE KNOW
p = 0.99, M₀ = 113.16 g/mol
FIND
degree of polymerization (Xₙ), number-average molar mass (Mn)
Step 1 · Find degree of polymerization (Xₙ)
Use Xₙ = 1 ÷ (1 − p)
One
chain
is
left
for
every
end
group
still
unreacted,
so
divide
by
1
−
p
.
Xₙ
=
1
1 − 0.99
Xₙ
=
1
0.01
Xₙ
=
100
→
Step
growth
needs
p
very
close
to
1:
even
99%
conversion
makes
chains
of
only
100
residues.
Step 2 · Find number-average molar mass (Mn)
Use Mn = Xₙ × M₀
Multiply the residues in the average chain by the molar mass of one residue.
Mn = 100 × 113.16
Mn = 11,316 g/mol
Check
Put the numbers back in. Both sides should match.
100
=
1
1 − 0.99
✓
11,316 = 100 × 113.16 ✓
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