A materials engineer computes the degree of polymerization for a batch of polymers. Given number-average molecular weight (Mn) = 195,500 g/mol; repeat unit molecular weight (M0) = 41.0000 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) = 195,500 g/mol, repeat unit molecular weight (M0) = 41.0000 g/mol.

4. Step 4 — Substitute the given values:

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

5. Step 5 — Evaluate:

   $$DP = 4768$$

6. Step 6 — Check: returning DP = 4,768 to

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

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

Why the other options are there

• 9,537 — kept a factor of two that cancels in the correct rearrangement.
• 2,384 — dropped that same factor in the other direction.
• 5,245 — 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) = 33.0000 g/mol; degree of polymerization (DP) = 4,455, 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) = 33.0000 g/mol, degree of polymerization (DP) = 4,455.

4. Step 4 — Substitute the given values:

   $$Mn = 4455 \times M_0$$

5. Step 5 — Evaluate:

   $$Mn = 147015\ \text{g/mol}$$

6. Step 6 — Check: returning Mn = 147,015 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

• 294,030 — kept a factor of two that cancels in the correct rearrangement.
• 73,508 — dropped that same factor in the other direction.
• 161,717 — 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) = 5,422, 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

   $$List the givens: number-average molecular weight (Mn) = 163,000 g/mol, degree of polymerization (DP) = 5,422$$

4. Step 4 — Substitute the given values:

   $$M_{0} = \dfrac{M_n}{5422}$$

5. Step 5 — Evaluate:

   $$M_{0} = 30.0627\ \text{g/mol}$$

6. Step 6 — Check: returning M0 = 30.0627 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

• 60.1254 — kept a factor of two that cancels in the correct rearrangement.
• 15.0314 — dropped that same factor in the other direction.
• 33.0690 — 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) = 164,500 g/mol; repeat unit molecular weight (M0) = 75.0000 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) = 164,500 g/mol, repeat unit molecular weight (M0) = 75.0000 g/mol.

4. Step 4 — Substitute the given values:

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

5. Step 5 — Evaluate:

   $$DP = 2193$$

6. Step 6 — Check: returning DP = 2,193 to

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

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

Why the other options are there

• 4,387 — kept a factor of two that cancels in the correct rearrangement.
• 1,097 — dropped that same factor in the other direction.
• 2,413 — 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) = 108.0 g/mol; degree of polymerization (DP) = 3,821, 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) = 108.0 g/mol, degree of polymerization (DP) = 3,821.

4. Step 4 — Substitute the given values:

   $$Mn = 3821 \times M_0$$

5. Step 5 — Evaluate:

   $$Mn = 412668\ \text{g/mol}$$

6. Step 6 — Check: returning Mn = 412,668 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

• 825,336 — kept a factor of two that cancels in the correct rearrangement.
• 206,334 — dropped that same factor in the other direction.
• 453,935 — 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) = 117,500 g/mol; degree of polymerization (DP) = 3,596, 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

   $$List the givens: number-average molecular weight (Mn) = 117,500 g/mol, degree of polymerization (DP) = 3,596$$

4. Step 4 — Substitute the given values:

   $$M_{0} = \dfrac{M_n}{3596}$$

5. Step 5 — Evaluate:

   $$M_{0} = 32.6752\ \text{g/mol}$$

6. Step 6 — Check: returning M0 = 32.6752 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

• 65.3504 — kept a factor of two that cancels in the correct rearrangement.
• 16.3376 — dropped that same factor in the other direction.
• 35.9427 — 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) = 108,000 g/mol; repeat unit molecular weight (M0) = 47.0000 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) = 108,000 g/mol, repeat unit molecular weight (M0) = 47.0000 g/mol.

4. Step 4 — Substitute the given values:

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

5. Step 5 — Evaluate:

   $$DP = 2298$$

6. Step 6 — Check: returning DP = 2,298 to

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

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

Why the other options are there

• 4,596 — kept a factor of two that cancels in the correct rearrangement.
• 1,149 — dropped that same factor in the other direction.
• 2,528 — 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) = 115.0 g/mol; degree of polymerization (DP) = 2,441, 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) = 115.0 g/mol, degree of polymerization (DP) = 2,441.

4. Step 4 — Substitute the given values:

   $$Mn = 2441 \times M_0$$

5. Step 5 — Evaluate:

   $$Mn = 280715\ \text{g/mol}$$

6. Step 6 — Check: returning Mn = 280,715 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

• 561,430 — kept a factor of two that cancels in the correct rearrangement.
• 140,358 — dropped that same factor in the other direction.
• 308,787 — 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) = 152,000 g/mol; degree of polymerization (DP) = 5,583, 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

   $$List the givens: number-average molecular weight (Mn) = 152,000 g/mol, degree of polymerization (DP) = 5,583$$

4. Step 4 — Substitute the given values:

   $$M_{0} = \dfrac{M_n}{5583}$$

5. Step 5 — Evaluate:

   $$M_{0} = 27.2255\ \text{g/mol}$$

6. Step 6 — Check: returning M0 = 27.2255 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

• 54.4510 — kept a factor of two that cancels in the correct rearrangement.
• 13.6128 — dropped that same factor in the other direction.
• 29.9481 — 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) = 137,500 g/mol; repeat unit molecular weight (M0) = 119.0 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) = 137,500 g/mol, repeat unit molecular weight (M0) = 119.0 g/mol.

4. Step 4 — Substitute the given values:

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

5. Step 5 — Evaluate:

   $$DP = 1155$$

6. Step 6 — Check: returning DP = 1,155 to

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

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

Why the other options are there

• 2,311 — kept a factor of two that cancels in the correct rearrangement.
• 577.7 — dropped that same factor in the other direction.
• 1,271 — rounded an intermediate value before the final step.

Reference: FE Handbook — Polymers