half-life of a radioactive isotope used in dose calculations Given decay constant (lambda) = 0.1021 1/yr, determine the half-life (t_half) in yr.

Start with the thinking

• The governing relation printed in this handbook section is Half-Life.
• Everything except t_half is given, so isolate t_half 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.
• Half-life relates the radioactive decay constant to the time required for activity to decrease to one-half its initial value.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$t_{1/2} = \dfrac{0.693}{\lambda}$$

2. Step 2 — Rearrange symbolically for t_half:

   $$t_{half} = \dfrac{0.693}{\lambda}$$

3. Step 3

   $$List the givens: decay constant (lambda) = 0.1021 1/yr$$

4. Step 4 — Substitute the given values:

   $$t_{half} = \dfrac{0.693}{0.1021}$$

5. Step 5 — Evaluate:

   $$t_{half} = 6.7875\ \text{yr}$$

6. Step 6 — Check: returning t_half = 6.7875 yr to

   $$t_{1/2} = \dfrac{0.693}{\lambda}$$

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

Why the other options are there

• 13.5749 — kept a factor of two that cancels in the correct rearrangement.
• 3.3937 — dropped that same factor in the other direction.
• 7.4662 — rounded an intermediate value before the final step.

Reference: FE Handbook — Half-Life

half-life determination from a measured decay constant Given half-life (t_half) = 5,025 yr, determine the decay constant (lambda) in 1/yr.

Start with the thinking

• The governing relation printed in this handbook section is Half-Life.
• Everything except lambda is given, so isolate lambda 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.
• Half-life relates the radioactive decay constant to the time required for activity to decrease to one-half its initial value.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$t_{1/2} = \dfrac{0.693}{\lambda}$$

2. Step 2 — Rearrange symbolically for lambda:

   $$\lambda = \dfrac{0.693}{t_{1/2}}$$

3. Step 3

   $$List the givens: half-life (t_half) = 5,025 yr$$

4. Step 4 — Substitute the given values:

   $$\lambda = \dfrac{0.693}{t_{1/2}}$$

5. Step 5 — Evaluate:

   $$\lambda = 0.0001\ \text{1/yr}$$

6. Step 6 — Check: returning lambda = 0.0001 1/yr to

   $$t_{1/2} = \dfrac{0.693}{\lambda}$$

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

Why the other options are there

• 0.0003 — kept a factor of two that cancels in the correct rearrangement.
• 0.0001 — dropped that same factor in the other direction.
• 0.0002 — rounded an intermediate value before the final step.

Reference: FE Handbook — Half-Life

radioactive half-life used to plan waste storage duration Given decay constant (lambda) = 0.4755 1/yr, determine the half-life (t_half) in yr.

Start with the thinking

• The governing relation printed in this handbook section is Half-Life.
• Everything except t_half is given, so isolate t_half 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.
• Half-life relates the radioactive decay constant to the time required for activity to decrease to one-half its initial value.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$t_{1/2} = \dfrac{0.693}{\lambda}$$

2. Step 2 — Rearrange symbolically for t_half:

   $$t_{half} = \dfrac{0.693}{\lambda}$$

3. Step 3

   $$List the givens: decay constant (lambda) = 0.4755 1/yr$$

4. Step 4 — Substitute the given values:

   $$t_{half} = \dfrac{0.693}{0.4755}$$

5. Step 5 — Evaluate:

   $$t_{half} = 1.4574\ \text{yr}$$

6. Step 6 — Check: returning t_half = 1.4574 yr to

   $$t_{1/2} = \dfrac{0.693}{\lambda}$$

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

Why the other options are there

• 2.9148 — kept a factor of two that cancels in the correct rearrangement.
• 0.7287 — dropped that same factor in the other direction.
• 1.6032 — rounded an intermediate value before the final step.

Reference: FE Handbook — Half-Life

half-life of a radioactive isotope used in dose calculations Given half-life (t_half) = 2,268 yr, determine the decay constant (lambda) in 1/yr.

Start with the thinking

• The governing relation printed in this handbook section is Half-Life.
• Everything except lambda is given, so isolate lambda 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.
• Half-life relates the radioactive decay constant to the time required for activity to decrease to one-half its initial value.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$t_{1/2} = \dfrac{0.693}{\lambda}$$

2. Step 2 — Rearrange symbolically for lambda:

   $$\lambda = \dfrac{0.693}{t_{1/2}}$$

3. Step 3

   $$List the givens: half-life (t_half) = 2,268 yr$$

4. Step 4 — Substitute the given values:

   $$\lambda = \dfrac{0.693}{t_{1/2}}$$

5. Step 5 — Evaluate:

   $$\lambda = 0.0003\ \text{1/yr}$$

6. Step 6 — Check: returning lambda = 0.0003 1/yr to

   $$t_{1/2} = \dfrac{0.693}{\lambda}$$

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

Why the other options are there

• 0.0006 — kept a factor of two that cancels in the correct rearrangement.
• 0.0002 — dropped that same factor in the other direction.
• 0.0003 — rounded an intermediate value before the final step.

Reference: FE Handbook — Half-Life

half-life determination from a measured decay constant Given decay constant (lambda) = 0.4167 1/yr, determine the half-life (t_half) in yr.

Start with the thinking

• The governing relation printed in this handbook section is Half-Life.
• Everything except t_half is given, so isolate t_half 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.
• Half-life relates the radioactive decay constant to the time required for activity to decrease to one-half its initial value.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$t_{1/2} = \dfrac{0.693}{\lambda}$$

2. Step 2 — Rearrange symbolically for t_half:

   $$t_{half} = \dfrac{0.693}{\lambda}$$

3. Step 3

   $$List the givens: decay constant (lambda) = 0.4167 1/yr$$

4. Step 4 — Substitute the given values:

   $$t_{half} = \dfrac{0.693}{0.4167}$$

5. Step 5 — Evaluate:

   $$t_{half} = 1.6631\ \text{yr}$$

6. Step 6 — Check: returning t_half = 1.6631 yr to

   $$t_{1/2} = \dfrac{0.693}{\lambda}$$

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

Why the other options are there

• 3.3261 — kept a factor of two that cancels in the correct rearrangement.
• 0.8315 — dropped that same factor in the other direction.
• 1.8294 — rounded an intermediate value before the final step.

Reference: FE Handbook — Half-Life

radioactive half-life used to plan waste storage duration Given half-life (t_half) = 3,871 yr, determine the decay constant (lambda) in 1/yr.

Start with the thinking

• The governing relation printed in this handbook section is Half-Life.
• Everything except lambda is given, so isolate lambda 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.
• Half-life relates the radioactive decay constant to the time required for activity to decrease to one-half its initial value.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$t_{1/2} = \dfrac{0.693}{\lambda}$$

2. Step 2 — Rearrange symbolically for lambda:

   $$\lambda = \dfrac{0.693}{t_{1/2}}$$

3. Step 3

   $$List the givens: half-life (t_half) = 3,871 yr$$

4. Step 4 — Substitute the given values:

   $$\lambda = \dfrac{0.693}{t_{1/2}}$$

5. Step 5 — Evaluate:

   $$\lambda = 0.0002\ \text{1/yr}$$

6. Step 6 — Check: returning lambda = 0.0002 1/yr to

   $$t_{1/2} = \dfrac{0.693}{\lambda}$$

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

Why the other options are there

• 0.0004 — kept a factor of two that cancels in the correct rearrangement.
• 0.0001 — dropped that same factor in the other direction.
• 0.0002 — rounded an intermediate value before the final step.

Reference: FE Handbook — Half-Life

half-life of a radioactive isotope used in dose calculations Given decay constant (lambda) = 0.4657 1/yr, determine the half-life (t_half) in yr.

Start with the thinking

• The governing relation printed in this handbook section is Half-Life.
• Everything except t_half is given, so isolate t_half 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.
• Half-life relates the radioactive decay constant to the time required for activity to decrease to one-half its initial value.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$t_{1/2} = \dfrac{0.693}{\lambda}$$

2. Step 2 — Rearrange symbolically for t_half:

   $$t_{half} = \dfrac{0.693}{\lambda}$$

3. Step 3

   $$List the givens: decay constant (lambda) = 0.4657 1/yr$$

4. Step 4 — Substitute the given values:

   $$t_{half} = \dfrac{0.693}{0.4657}$$

5. Step 5 — Evaluate:

   $$t_{half} = 1.4881\ \text{yr}$$

6. Step 6 — Check: returning t_half = 1.4881 yr to

   $$t_{1/2} = \dfrac{0.693}{\lambda}$$

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

Why the other options are there

• 2.9762 — kept a factor of two that cancels in the correct rearrangement.
• 0.7440 — dropped that same factor in the other direction.
• 1.6369 — rounded an intermediate value before the final step.

Reference: FE Handbook — Half-Life

half-life determination from a measured decay constant Given half-life (t_half) = 6,243 yr, determine the decay constant (lambda) in 1/yr.

Start with the thinking

• The governing relation printed in this handbook section is Half-Life.
• Everything except lambda is given, so isolate lambda 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.
• Half-life relates the radioactive decay constant to the time required for activity to decrease to one-half its initial value.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$t_{1/2} = \dfrac{0.693}{\lambda}$$

2. Step 2 — Rearrange symbolically for lambda:

   $$\lambda = \dfrac{0.693}{t_{1/2}}$$

3. Step 3

   $$List the givens: half-life (t_half) = 6,243 yr$$

4. Step 4 — Substitute the given values:

   $$\lambda = \dfrac{0.693}{t_{1/2}}$$

5. Step 5 — Evaluate:

   $$\lambda = 0.0001\ \text{1/yr}$$

6. Step 6 — Check: returning lambda = 0.0001 1/yr to

   $$t_{1/2} = \dfrac{0.693}{\lambda}$$

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

Why the other options are there

• 0.0002 — kept a factor of two that cancels in the correct rearrangement.
• 0.0001 — dropped that same factor in the other direction.
• 0.0001 — rounded an intermediate value before the final step.

Reference: FE Handbook — Half-Life

radioactive half-life used to plan waste storage duration Given decay constant (lambda) = 0.3575 1/yr, determine the half-life (t_half) in yr.

Start with the thinking

• The governing relation printed in this handbook section is Half-Life.
• Everything except t_half is given, so isolate t_half 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.
• Half-life relates the radioactive decay constant to the time required for activity to decrease to one-half its initial value.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$t_{1/2} = \dfrac{0.693}{\lambda}$$

2. Step 2 — Rearrange symbolically for t_half:

   $$t_{half} = \dfrac{0.693}{\lambda}$$

3. Step 3

   $$List the givens: decay constant (lambda) = 0.3575 1/yr$$

4. Step 4 — Substitute the given values:

   $$t_{half} = \dfrac{0.693}{0.3575}$$

5. Step 5 — Evaluate:

   $$t_{half} = 1.9385\ \text{yr}$$

6. Step 6 — Check: returning t_half = 1.9385 yr to

   $$t_{1/2} = \dfrac{0.693}{\lambda}$$

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

Why the other options are there

• 3.8769 — kept a factor of two that cancels in the correct rearrangement.
• 0.9692 — dropped that same factor in the other direction.
• 2.1323 — rounded an intermediate value before the final step.

Reference: FE Handbook — Half-Life

half-life of a radioactive isotope used in dose calculations Given half-life (t_half) = 91.4000 yr, determine the decay constant (lambda) in 1/yr.

Start with the thinking

• The governing relation printed in this handbook section is Half-Life.
• Everything except lambda is given, so isolate lambda 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.
• Half-life relates the radioactive decay constant to the time required for activity to decrease to one-half its initial value.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$t_{1/2} = \dfrac{0.693}{\lambda}$$

2. Step 2 — Rearrange symbolically for lambda:

   $$\lambda = \dfrac{0.693}{t_{1/2}}$$

3. Step 3

   $$List the givens: half-life (t_half) = 91.4000 yr$$

4. Step 4 — Substitute the given values:

   $$\lambda = \dfrac{0.693}{t_{1/2}}$$

5. Step 5 — Evaluate:

   $$\lambda = 0.0076\ \text{1/yr}$$

6. Step 6 — Check: returning lambda = 0.0076 1/yr to

   $$t_{1/2} = \dfrac{0.693}{\lambda}$$

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

Why the other options are there

• 0.0152 — kept a factor of two that cancels in the correct rearrangement.
• 0.0038 — dropped that same factor in the other direction.
• 0.0083 — rounded an intermediate value before the final step.

Reference: FE Handbook — Half-Life