A materials engineer computes the degree of polymerization for a batch of polymers. Given number-average molecular weight (Mn) = 113,000 g/mol; repeat unit molecular weight (M0) = 140.0 g/mol, determine the degree of polymerization (DP).

Start with the thinking

• The governing relation printed in this handbook section is Polymers — degree of polymerization.
• Everything except DP is given, so isolate DP symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Polymers are characterized by their degree of polymerization, the ratio of chain molecular weight to repeat unit weight.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$DP = \dfrac{M_n}{M_0}$$

2. Step 2 — Rearrange symbolically for DP:

   $$DP = \dfrac{M_n}{M_0}$$

3. Step 3 — List the givens: number-average molecular weight (Mn) = 113,000 g/mol, repeat unit molecular weight (M0) = 140.0 g/mol.

4. Step 4 — Substitute the given values:

   $$DP = \dfrac{M_n}{M_0}$$

5. Step 5 — Evaluate:

   $$DP = 807.1$$

6. Step 6 — Check: returning DP = 807.1 to

   $$DP = \dfrac{M_n}{M_0}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 1,614 — kept a factor of two that cancels in the correct rearrangement.
• 403.6 — dropped that same factor in the other direction.
• 887.9 — rounded an intermediate value before the final step.

Reference: FE Handbook — Polymers

A student relates molecular weight to the degree of polymerization for a polymer chain. Given repeat unit molecular weight (M0) = 147.0 g/mol; degree of polymerization (DP) = 2,286, determine the number-average molecular weight (Mn) in g/mol.

Start with the thinking

• The governing relation printed in this handbook section is Polymers — degree of polymerization.
• Everything except Mn is given, so isolate Mn symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Polymers are characterized by their degree of polymerization, the ratio of chain molecular weight to repeat unit weight.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$DP = \dfrac{M_n}{M_0}$$

2. Step 2 — Rearrange symbolically for Mn:

   $$Mn = DP \times M_0$$

3. Step 3 — List the givens: repeat unit molecular weight (M0) = 147.0 g/mol, degree of polymerization (DP) = 2,286.

4. Step 4 — Substitute the given values:

   $$Mn = 2286 \times M_0$$

5. Step 5 — Evaluate:

   $$Mn = 336042\ \text{g/mol}$$

6. Step 6 — Check: returning Mn = 336,042 g/mol to

   $$DP = \dfrac{M_n}{M_0}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 672,084 — kept a factor of two that cancels in the correct rearrangement.
• 168,021 — dropped that same factor in the other direction.
• 369,646 — rounded an intermediate value before the final step.

Reference: FE Handbook — Polymers

The polymer's degree of polymerization influences its mechanical strength. Given number-average molecular weight (Mn) = 163,000 g/mol; degree of polymerization (DP) = 3,984, determine the repeat unit molecular weight (M0) in g/mol.

Start with the thinking

• The governing relation printed in this handbook section is Polymers — degree of polymerization.
• Everything except M0 is given, so isolate M0 symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Polymers are characterized by their degree of polymerization, the ratio of chain molecular weight to repeat unit weight.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$DP = \dfrac{M_n}{M_0}$$

2. Step 2 — Rearrange symbolically for M0:

   $$M_{0} = \dfrac{M_n}{DP}$$

3. Step 3

   $$List the givens: number-average molecular weight (Mn) = 163,000 g/mol, degree of polymerization (DP) = 3,984$$

4. Step 4 — Substitute the given values:

   $$M_{0} = \dfrac{M_n}{3984}$$

5. Step 5 — Evaluate:

   $$M_{0} = 40.9137\ \text{g/mol}$$

6. Step 6 — Check: returning M0 = 40.9137 g/mol to

   $$DP = \dfrac{M_n}{M_0}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 81.8273 — kept a factor of two that cancels in the correct rearrangement.
• 20.4568 — dropped that same factor in the other direction.
• 45.0050 — rounded an intermediate value before the final step.

Reference: FE Handbook — Polymers

A materials engineer computes the degree of polymerization for a batch of polymers. Given number-average molecular weight (Mn) = 23,500 g/mol; repeat unit molecular weight (M0) = 60.0000 g/mol, determine the degree of polymerization (DP).

Start with the thinking

• The governing relation printed in this handbook section is Polymers — degree of polymerization.
• Everything except DP is given, so isolate DP symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Polymers are characterized by their degree of polymerization, the ratio of chain molecular weight to repeat unit weight.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$DP = \dfrac{M_n}{M_0}$$

2. Step 2 — Rearrange symbolically for DP:

   $$DP = \dfrac{M_n}{M_0}$$

3. Step 3 — List the givens: number-average molecular weight (Mn) = 23,500 g/mol, repeat unit molecular weight (M0) = 60.0000 g/mol.

4. Step 4 — Substitute the given values:

   $$DP = \dfrac{M_n}{M_0}$$

5. Step 5 — Evaluate:

   $$DP = 391.7$$

6. Step 6 — Check: returning DP = 391.7 to

   $$DP = \dfrac{M_n}{M_0}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 783.3 — kept a factor of two that cancels in the correct rearrangement.
• 195.8 — dropped that same factor in the other direction.
• 430.8 — rounded an intermediate value before the final step.

Reference: FE Handbook — Polymers

A student relates molecular weight to the degree of polymerization for a polymer chain. Given repeat unit molecular weight (M0) = 145.0 g/mol; degree of polymerization (DP) = 4,965, determine the number-average molecular weight (Mn) in g/mol.

Start with the thinking

• The governing relation printed in this handbook section is Polymers — degree of polymerization.
• Everything except Mn is given, so isolate Mn symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Polymers are characterized by their degree of polymerization, the ratio of chain molecular weight to repeat unit weight.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$DP = \dfrac{M_n}{M_0}$$

2. Step 2 — Rearrange symbolically for Mn:

   $$Mn = DP \times M_0$$

3. Step 3 — List the givens: repeat unit molecular weight (M0) = 145.0 g/mol, degree of polymerization (DP) = 4,965.

4. Step 4 — Substitute the given values:

   $$Mn = 4965 \times M_0$$

5. Step 5 — Evaluate:

   $$Mn = 719925\ \text{g/mol}$$

6. Step 6 — Check: returning Mn = 719,925 g/mol to

   $$DP = \dfrac{M_n}{M_0}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 1,439,850 — kept a factor of two that cancels in the correct rearrangement.
• 359,963 — dropped that same factor in the other direction.
• 791,918 — rounded an intermediate value before the final step.

Reference: FE Handbook — Polymers

The polymer's degree of polymerization influences its mechanical strength. Given number-average molecular weight (Mn) = 21,500 g/mol; degree of polymerization (DP) = 1,171, determine the repeat unit molecular weight (M0) in g/mol.

Start with the thinking

• The governing relation printed in this handbook section is Polymers — degree of polymerization.
• Everything except M0 is given, so isolate M0 symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Polymers are characterized by their degree of polymerization, the ratio of chain molecular weight to repeat unit weight.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$DP = \dfrac{M_n}{M_0}$$

2. Step 2 — Rearrange symbolically for M0:

   $$M_{0} = \dfrac{M_n}{DP}$$

3. Step 3

   $$List the givens: number-average molecular weight (Mn) = 21,500 g/mol, degree of polymerization (DP) = 1,171$$

4. Step 4 — Substitute the given values:

   $$M_{0} = \dfrac{M_n}{1171}$$

5. Step 5 — Evaluate:

   $$M_{0} = 18.3604\ \text{g/mol}$$

6. Step 6 — Check: returning M0 = 18.3604 g/mol to

   $$DP = \dfrac{M_n}{M_0}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 36.7208 — kept a factor of two that cancels in the correct rearrangement.
• 9.1802 — dropped that same factor in the other direction.
• 20.1964 — rounded an intermediate value before the final step.

Reference: FE Handbook — Polymers

A materials engineer computes the degree of polymerization for a batch of polymers. Given number-average molecular weight (Mn) = 146,500 g/mol; repeat unit molecular weight (M0) = 70.0000 g/mol, determine the degree of polymerization (DP).

Start with the thinking

• The governing relation printed in this handbook section is Polymers — degree of polymerization.
• Everything except DP is given, so isolate DP symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Polymers are characterized by their degree of polymerization, the ratio of chain molecular weight to repeat unit weight.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$DP = \dfrac{M_n}{M_0}$$

2. Step 2 — Rearrange symbolically for DP:

   $$DP = \dfrac{M_n}{M_0}$$

3. Step 3 — List the givens: number-average molecular weight (Mn) = 146,500 g/mol, repeat unit molecular weight (M0) = 70.0000 g/mol.

4. Step 4 — Substitute the given values:

   $$DP = \dfrac{M_n}{M_0}$$

5. Step 5 — Evaluate:

   $$DP = 2093$$

6. Step 6 — Check: returning DP = 2,093 to

   $$DP = \dfrac{M_n}{M_0}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 4,186 — kept a factor of two that cancels in the correct rearrangement.
• 1,046 — dropped that same factor in the other direction.
• 2,302 — rounded an intermediate value before the final step.

Reference: FE Handbook — Polymers

A student relates molecular weight to the degree of polymerization for a polymer chain. Given repeat unit molecular weight (M0) = 42.0000 g/mol; degree of polymerization (DP) = 5,711, determine the number-average molecular weight (Mn) in g/mol.

Start with the thinking

• The governing relation printed in this handbook section is Polymers — degree of polymerization.
• Everything except Mn is given, so isolate Mn symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Polymers are characterized by their degree of polymerization, the ratio of chain molecular weight to repeat unit weight.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$DP = \dfrac{M_n}{M_0}$$

2. Step 2 — Rearrange symbolically for Mn:

   $$Mn = DP \times M_0$$

3. Step 3 — List the givens: repeat unit molecular weight (M0) = 42.0000 g/mol, degree of polymerization (DP) = 5,711.

4. Step 4 — Substitute the given values:

   $$Mn = 5711 \times M_0$$

5. Step 5 — Evaluate:

   $$Mn = 239862\ \text{g/mol}$$

6. Step 6 — Check: returning Mn = 239,862 g/mol to

   $$DP = \dfrac{M_n}{M_0}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 479,724 — kept a factor of two that cancels in the correct rearrangement.
• 119,931 — dropped that same factor in the other direction.
• 263,848 — rounded an intermediate value before the final step.

Reference: FE Handbook — Polymers

The polymer's degree of polymerization influences its mechanical strength. Given number-average molecular weight (Mn) = 129,000 g/mol; degree of polymerization (DP) = 4,522, determine the repeat unit molecular weight (M0) in g/mol.

Start with the thinking

• The governing relation printed in this handbook section is Polymers — degree of polymerization.
• Everything except M0 is given, so isolate M0 symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Polymers are characterized by their degree of polymerization, the ratio of chain molecular weight to repeat unit weight.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$DP = \dfrac{M_n}{M_0}$$

2. Step 2 — Rearrange symbolically for M0:

   $$M_{0} = \dfrac{M_n}{DP}$$

3. Step 3

   $$List the givens: number-average molecular weight (Mn) = 129,000 g/mol, degree of polymerization (DP) = 4,522$$

4. Step 4 — Substitute the given values:

   $$M_{0} = \dfrac{M_n}{4522}$$

5. Step 5 — Evaluate:

   $$M_{0} = 28.5272\ \text{g/mol}$$

6. Step 6 — Check: returning M0 = 28.5272 g/mol to

   $$DP = \dfrac{M_n}{M_0}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 57.0544 — kept a factor of two that cancels in the correct rearrangement.
• 14.2636 — dropped that same factor in the other direction.
• 31.3799 — rounded an intermediate value before the final step.

Reference: FE Handbook — Polymers

A materials engineer computes the degree of polymerization for a batch of polymers. Given number-average molecular weight (Mn) = 136,000 g/mol; repeat unit molecular weight (M0) = 78.0000 g/mol, determine the degree of polymerization (DP).

Start with the thinking

• The governing relation printed in this handbook section is Polymers — degree of polymerization.
• Everything except DP is given, so isolate DP symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Polymers are characterized by their degree of polymerization, the ratio of chain molecular weight to repeat unit weight.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$DP = \dfrac{M_n}{M_0}$$

2. Step 2 — Rearrange symbolically for DP:

   $$DP = \dfrac{M_n}{M_0}$$

3. Step 3 — List the givens: number-average molecular weight (Mn) = 136,000 g/mol, repeat unit molecular weight (M0) = 78.0000 g/mol.

4. Step 4 — Substitute the given values:

   $$DP = \dfrac{M_n}{M_0}$$

5. Step 5 — Evaluate:

   $$DP = 1744$$

6. Step 6 — Check: returning DP = 1,744 to

   $$DP = \dfrac{M_n}{M_0}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 3,487 — kept a factor of two that cancels in the correct rearrangement.
• 871.8 — dropped that same factor in the other direction.
• 1,918 — rounded an intermediate value before the final step.

Reference: FE Handbook — Polymers