SME
Biomaterials · Problem type
Molecular weight and dispersity
Use this for “A PLGA sample has 10 chains of 10 kDa, 20 of 20 kDa and 10 of 40 kDa. Find Mn, Mw and the dispersity.”
Chart
10
20
10
This is an example. Type your own values to start a new problem.
N₁
Chains at M₁
example
N₂
Chains at M₂
example
N₃
Chains at M₃
example
M₁
Chain mass 1
example · kDa
M₂
Chain mass 2
example · kDa
M₃
Chain mass 3
example · kDa
Mₙ
Number average
calculated · kDa
Mw
Weight average
calculated · kDa
Đ
Dispersity
calculated
Formulas
Mₙ
=
(
N₁
×
M₁
+
N₂
×
M₂
+
N₃
×
M₃
)
÷
(
N₁
+
N₂
+
N₃
)
22.5 = (10 × 10 + 20 × 20 + 10 × 40) ÷ (10 + 20 + 10)
Mw
=
(
N₁
×
M₁
2
+
N₂
×
M₂
2
+
N₃
×
M₃
2
)
÷
(
N₁
×
M₁
+
N₂
×
M₂
+
N₃
×
M₃
)
27.78
=
(10
×
10
2
+
20
×
20
2
+
10
×
40
2
)
÷
(10
×
10
+
20
×
20
+
10
×
40)
Đ
=
Mw
Mₙ
1.235
=
27.78
22.5
Assumptions
•
N counts the chains (or moles of chains) of each length; M is the molar mass of one chain.
•
Mₙ weighs every chain the same; Mw weighs each by its mass, so long chains count more.
•
Đ = Mw ÷ Mₙ is 1 when every chain is the same length and grows as lengths spread.
Step-by-step
WE KNOW
N₁ = 10, M₁ = 10 kDa, N₂ = 20, M₂ = 20 kDa, N₃ = 10, M₃ = 40 kDa
FIND
number average (Mₙ), weight average (Mw), dispersity (Đ)
Step 1 · Find number average (Mₙ)
Use Mₙ = (N₁ × M₁ + N₂ × M₂ + N₃ × M₃) ÷ (N₁ + N₂ + N₃)
The total mass of all the chains over how many chains there are.
Mₙ = (10 × 10 + 20 × 20 + 10 × 40) ÷ (10 + 20 + 10)
Mₙ
=
100 + 400 + 400
30 + 10
Mₙ
=
500 + 400
40
Mₙ
=
900
40
Mₙ = 22.5 kDa
Step 2 · Find weight average (Mw)
Use Mw = (N₁ × M₁² + N₂ × M₂² + N₃ × M₃²) ÷ (N₁ × M₁ + N₂ × M₂ + N₃ × M₃)
Average each chain’s mass weighted by the mass it brings: ΣNM² over the total mass ΣNM.
Mw
=
(10
×
10
2
+
20
×
20
2
+
10
×
40
2
)
÷
(10
×
10
+
20
×
20
+
10
×
40)
Mw = (10 × 100 + 20 × 400 + 10 × 1,600) ÷ (100 + 400 + 400)
Mw
=
1,000 + 8,000 + 16,000
500 + 400
Mw
=
9,000 + 16,000
900
Mw
=
25,000
900
Mw = 27.78 kDa
Step 3 · Find dispersity (Đ)
Use Đ = Mw ÷ Mₙ
Dispersity compares the two averages; the wider the spread, the larger Mw gets.
Đ
=
27.78
22.5
Đ
=
1.235
Check
Put the numbers back in. Both sides should match.
22.5 = (10 × 10 + 20 × 20 + 10 × 40) ÷ (10 + 20 + 10) ✓
27.78
=
(10
×
10
2
+
20
×
20
2
+
10
×
40
2
)
÷
(10
×
10
+
20
×
20
+
10
×
40)
✓
1.235
=
27.78
22.5
✓
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