A metallurgist calculates the diffusion coefficient of carbon in steel at a carburizing temperature. Given pre-exponential factor (D0) = 0.0001 m^2/s; activation energy (Q) = 114,000 J/mol; gas constant (R) = 8.3140 J/(mol K); temperature (T) = 1,140 K, determine the diffusion coefficient (D) in m^2/s.

Start with the thinking

• The governing relation printed in this handbook section is Diffusion coefficient — Arrhenius form.
• Everything except D is given, so isolate D 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.
• The diffusion coefficient of atoms in a solid follows an Arrhenius temperature dependence.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$D = D_0 e^{-Q/(RT)}$$

2. Step 2 — Rearrange symbolically for D:

   $$D = D_0 e^{-Q/(RT)}$$

3. Step 3 — List the givens: pre-exponential factor (D0) = 0.0001 m^2/s, activation energy (Q) = 114,000 J/mol, gas constant (R) = 8.3140 J/(mol K), temperature (T) = 1,140 K.

4. Step 4 — Substitute the given values:

   $$D = D_0 e^{-114000/(R1140)}$$

5. Step 5 — Evaluate:

   D = 0.0000\ \text{m^2/s}

6. Step 6 — Check: returning D = 0.0000 m^2/s to

   $$D = D_0 e^{-Q/(RT)}$$

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

Why the other options are there

• 0.0000 — kept a factor of two that cancels in the correct rearrangement.
• 0.0000 — dropped that same factor in the other direction.
• 0.0000 — rounded an intermediate value before the final step.

Reference: FE Handbook — Diffusion Coefficient

The diffusion coefficient rises sharply as temperature increases during processing. Given pre-exponential factor (D0) = 0.0001 m^2/s; activation energy (Q) = 125,000 J/mol; gas constant (R) = 8.3140 J/(mol K); diffusion coefficient (D) = 0.0000 m^2/s, determine the temperature (T) in K.

Start with the thinking

• The governing relation printed in this handbook section is Diffusion coefficient — Arrhenius form.
• Everything except T is given, so isolate T 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.
• The diffusion coefficient of atoms in a solid follows an Arrhenius temperature dependence.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$D = D_0 e^{-Q/(RT)}$$

2. Step 2 — Rearrange symbolically for T:

   $$T = \dfrac{-Q}{R\ln(D/D_0)}$$

3. Step 3 — List the givens: pre-exponential factor (D0) = 0.0001 m^2/s, activation energy (Q) = 125,000 J/mol, gas constant (R) = 8.3140 J/(mol K), diffusion coefficient (D) = 0.0000 m^2/s.

4. Step 4 — Substitute the given values:

   $$T = \dfrac{-125000}{8.3140\ln(0.0000/D_0)}$$

5. Step 5 — Evaluate:

   $$T = 3021\ \text{K}$$

6. Step 6 — Check: returning T = 3,021 K to

   $$D = D_0 e^{-Q/(RT)}$$

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

Why the other options are there

• 6,042 — kept a factor of two that cancels in the correct rearrangement.
• 1,511 — dropped that same factor in the other direction.
• 3,323 — rounded an intermediate value before the final step.

Reference: FE Handbook — Diffusion Coefficient

A metallurgist calculates the diffusion coefficient of carbon in steel at a carburizing temperature. Given pre-exponential factor (D0) = 0.0003 m^2/s; activation energy (Q) = 191,000 J/mol; gas constant (R) = 8.3140 J/(mol K); temperature (T) = 900.0 K, determine the diffusion coefficient (D) in m^2/s.

Start with the thinking

• The governing relation printed in this handbook section is Diffusion coefficient — Arrhenius form.
• Everything except D is given, so isolate D 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.
• The diffusion coefficient of atoms in a solid follows an Arrhenius temperature dependence.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$D = D_0 e^{-Q/(RT)}$$

2. Step 2 — Rearrange symbolically for D:

   $$D = D_0 e^{-Q/(RT)}$$

3. Step 3 — List the givens: pre-exponential factor (D0) = 0.0003 m^2/s, activation energy (Q) = 191,000 J/mol, gas constant (R) = 8.3140 J/(mol K), temperature (T) = 900.0 K.

4. Step 4 — Substitute the given values:

   $$D = D_0 e^{-191000/(R900.0)}$$

5. Step 5 — Evaluate:

   D = 0.0000\ \text{m^2/s}

6. Step 6 — Check: returning D = 0.0000 m^2/s to

   $$D = D_0 e^{-Q/(RT)}$$

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

Why the other options are there

• 0.0000 — kept a factor of two that cancels in the correct rearrangement.
• 0.0000 — dropped that same factor in the other direction.
• 0.0000 — rounded an intermediate value before the final step.

Reference: FE Handbook — Diffusion Coefficient

The diffusion coefficient rises sharply as temperature increases during processing. Given pre-exponential factor (D0) = 0.0001 m^2/s; activation energy (Q) = 172,000 J/mol; gas constant (R) = 8.3140 J/(mol K); diffusion coefficient (D) = 0.0000 m^2/s, determine the temperature (T) in K.

Start with the thinking

• The governing relation printed in this handbook section is Diffusion coefficient — Arrhenius form.
• Everything except T is given, so isolate T 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.
• The diffusion coefficient of atoms in a solid follows an Arrhenius temperature dependence.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$D = D_0 e^{-Q/(RT)}$$

2. Step 2 — Rearrange symbolically for T:

   $$T = \dfrac{-Q}{R\ln(D/D_0)}$$

3. Step 3 — List the givens: pre-exponential factor (D0) = 0.0001 m^2/s, activation energy (Q) = 172,000 J/mol, gas constant (R) = 8.3140 J/(mol K), diffusion coefficient (D) = 0.0000 m^2/s.

4. Step 4 — Substitute the given values:

   $$T = \dfrac{-172000}{8.3140\ln(0.0000/D_0)}$$

5. Step 5 — Evaluate:

   $$T = 0.0000\ \text{K}$$

6. Step 6 — Check: returning T = 0.0000 K to

   $$D = D_0 e^{-Q/(RT)}$$

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

Why the other options are there

• 0.0000 — kept a factor of two that cancels in the correct rearrangement.
• 0.0000 — dropped that same factor in the other direction.
• 0.0000 — rounded an intermediate value before the final step.

Reference: FE Handbook — Diffusion Coefficient

A metallurgist calculates the diffusion coefficient of carbon in steel at a carburizing temperature. Given pre-exponential factor (D0) = 0.0004 m^2/s; activation energy (Q) = 164,000 J/mol; gas constant (R) = 8.3140 J/(mol K); temperature (T) = 720.0 K, determine the diffusion coefficient (D) in m^2/s.

Start with the thinking

• The governing relation printed in this handbook section is Diffusion coefficient — Arrhenius form.
• Everything except D is given, so isolate D 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.
• The diffusion coefficient of atoms in a solid follows an Arrhenius temperature dependence.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$D = D_0 e^{-Q/(RT)}$$

2. Step 2 — Rearrange symbolically for D:

   $$D = D_0 e^{-Q/(RT)}$$

3. Step 3 — List the givens: pre-exponential factor (D0) = 0.0004 m^2/s, activation energy (Q) = 164,000 J/mol, gas constant (R) = 8.3140 J/(mol K), temperature (T) = 720.0 K.

4. Step 4 — Substitute the given values:

   $$D = D_0 e^{-164000/(R720.0)}$$

5. Step 5 — Evaluate:

   D = 0.0000\ \text{m^2/s}

6. Step 6 — Check: returning D = 0.0000 m^2/s to

   $$D = D_0 e^{-Q/(RT)}$$

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

Why the other options are there

• 0.0000 — kept a factor of two that cancels in the correct rearrangement.
• 0.0000 — dropped that same factor in the other direction.
• 0.0000 — rounded an intermediate value before the final step.

Reference: FE Handbook — Diffusion Coefficient

An engineer estimates the diffusion coefficient for a heat-treated alloy. Given pre-exponential factor (D0) = 0.0001 m^2/s; gas constant (R) = 8.3140 J/(mol K); temperature (T) = 925.0 K; diffusion coefficient (D) = 0.0000 m^2/s, determine the activation energy (Q) in J/mol.

Start with the thinking

• The governing relation printed in this handbook section is Diffusion coefficient — Arrhenius form.
• Everything except Q is given, so isolate Q 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.
• The diffusion coefficient of atoms in a solid follows an Arrhenius temperature dependence.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$D = D_0 e^{-Q/(RT)}$$

2. Step 2 — Rearrange symbolically for Q:

   $$Q = -RT\ln(D/D_0)$$

3. Step 3 — List the givens: pre-exponential factor (D0) = 0.0001 m^2/s, gas constant (R) = 8.3140 J/(mol K), temperature (T) = 925.0 K, diffusion coefficient (D) = 0.0000 m^2/s.

4. Step 4 — Substitute the given values:

   $$Q = -R925.0\ln(0.0000/D_0)$$

5. Step 5 — Evaluate:

   $$Q = 35021\ \text{J/mol}$$

6. Step 6 — Check: returning Q = 35,021 J/mol to

   $$D = D_0 e^{-Q/(RT)}$$

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

Why the other options are there

• 70,043 — kept a factor of two that cancels in the correct rearrangement.
• 17,511 — dropped that same factor in the other direction.
• 38,523 — rounded an intermediate value before the final step.

Reference: FE Handbook — Diffusion Coefficient

The diffusion coefficient rises sharply as temperature increases during processing. Given pre-exponential factor (D0) = 0.0001 m^2/s; activation energy (Q) = 198,000 J/mol; gas constant (R) = 8.3140 J/(mol K); diffusion coefficient (D) = 0.0000 m^2/s, determine the temperature (T) in K.

Start with the thinking

• The governing relation printed in this handbook section is Diffusion coefficient — Arrhenius form.
• Everything except T is given, so isolate T 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.
• The diffusion coefficient of atoms in a solid follows an Arrhenius temperature dependence.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$D = D_0 e^{-Q/(RT)}$$

2. Step 2 — Rearrange symbolically for T:

   $$T = \dfrac{-Q}{R\ln(D/D_0)}$$

3. Step 3 — List the givens: pre-exponential factor (D0) = 0.0001 m^2/s, activation energy (Q) = 198,000 J/mol, gas constant (R) = 8.3140 J/(mol K), diffusion coefficient (D) = 0.0000 m^2/s.

4. Step 4 — Substitute the given values:

   $$T = \dfrac{-198000}{8.3140\ln(0.0000/D_0)}$$

5. Step 5 — Evaluate:

   $$T = 0.0000\ \text{K}$$

6. Step 6 — Check: returning T = 0.0000 K to

   $$D = D_0 e^{-Q/(RT)}$$

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

Why the other options are there

• 0.0000 — kept a factor of two that cancels in the correct rearrangement.
• 0.0000 — dropped that same factor in the other direction.
• 0.0000 — rounded an intermediate value before the final step.

Reference: FE Handbook — Diffusion Coefficient

A metallurgist calculates the diffusion coefficient of carbon in steel at a carburizing temperature. Given pre-exponential factor (D0) = 0.0001 m^2/s; activation energy (Q) = 122,000 J/mol; gas constant (R) = 8.3140 J/(mol K); temperature (T) = 940.0 K, determine the diffusion coefficient (D) in m^2/s.

Start with the thinking

• The governing relation printed in this handbook section is Diffusion coefficient — Arrhenius form.
• Everything except D is given, so isolate D 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.
• The diffusion coefficient of atoms in a solid follows an Arrhenius temperature dependence.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$D = D_0 e^{-Q/(RT)}$$

2. Step 2 — Rearrange symbolically for D:

   $$D = D_0 e^{-Q/(RT)}$$

3. Step 3 — List the givens: pre-exponential factor (D0) = 0.0001 m^2/s, activation energy (Q) = 122,000 J/mol, gas constant (R) = 8.3140 J/(mol K), temperature (T) = 940.0 K.

4. Step 4 — Substitute the given values:

   $$D = D_0 e^{-122000/(R940.0)}$$

5. Step 5 — Evaluate:

   D = 0.0000\ \text{m^2/s}

6. Step 6 — Check: returning D = 0.0000 m^2/s to

   $$D = D_0 e^{-Q/(RT)}$$

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

Why the other options are there

• 0.0000 — kept a factor of two that cancels in the correct rearrangement.
• 0.0000 — dropped that same factor in the other direction.
• 0.0000 — rounded an intermediate value before the final step.

Reference: FE Handbook — Diffusion Coefficient

An engineer estimates the diffusion coefficient for a heat-treated alloy. Given pre-exponential factor (D0) = 0.0004 m^2/s; gas constant (R) = 8.3140 J/(mol K); temperature (T) = 755.0 K; diffusion coefficient (D) = 0.0000 m^2/s, determine the activation energy (Q) in J/mol.

Start with the thinking

• The governing relation printed in this handbook section is Diffusion coefficient — Arrhenius form.
• Everything except Q is given, so isolate Q 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.
• The diffusion coefficient of atoms in a solid follows an Arrhenius temperature dependence.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$D = D_0 e^{-Q/(RT)}$$

2. Step 2 — Rearrange symbolically for Q:

   $$Q = -RT\ln(D/D_0)$$

3. Step 3 — List the givens: pre-exponential factor (D0) = 0.0004 m^2/s, gas constant (R) = 8.3140 J/(mol K), temperature (T) = 755.0 K, diffusion coefficient (D) = 0.0000 m^2/s.

4. Step 4 — Substitute the given values:

   $$Q = -R755.0\ln(0.0000/D_0)$$

5. Step 5 — Evaluate:

   $$Q = 38193\ \text{J/mol}$$

6. Step 6 — Check: returning Q = 38,193 J/mol to

   $$D = D_0 e^{-Q/(RT)}$$

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

Why the other options are there

• 76,386 — kept a factor of two that cancels in the correct rearrangement.
• 19,096 — dropped that same factor in the other direction.
• 42,012 — rounded an intermediate value before the final step.

Reference: FE Handbook — Diffusion Coefficient

The diffusion coefficient rises sharply as temperature increases during processing. Given pre-exponential factor (D0) = 0.0001 m^2/s; activation energy (Q) = 168,000 J/mol; gas constant (R) = 8.3140 J/(mol K); diffusion coefficient (D) = 0.0000 m^2/s, determine the temperature (T) in K.

Start with the thinking

• The governing relation printed in this handbook section is Diffusion coefficient — Arrhenius form.
• Everything except T is given, so isolate T 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.
• The diffusion coefficient of atoms in a solid follows an Arrhenius temperature dependence.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$D = D_0 e^{-Q/(RT)}$$

2. Step 2 — Rearrange symbolically for T:

   $$T = \dfrac{-Q}{R\ln(D/D_0)}$$

3. Step 3 — List the givens: pre-exponential factor (D0) = 0.0001 m^2/s, activation energy (Q) = 168,000 J/mol, gas constant (R) = 8.3140 J/(mol K), diffusion coefficient (D) = 0.0000 m^2/s.

4. Step 4 — Substitute the given values:

   $$T = \dfrac{-168000}{8.3140\ln(0.0000/D_0)}$$

5. Step 5 — Evaluate:

   $$T = 4469\ \text{K}$$

6. Step 6 — Check: returning T = 4,469 K to

   $$D = D_0 e^{-Q/(RT)}$$

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

Why the other options are there

• 8,938 — kept a factor of two that cancels in the correct rearrangement.
• 2,234 — dropped that same factor in the other direction.
• 4,916 — rounded an intermediate value before the final step.

Reference: FE Handbook — Diffusion Coefficient