A wastewater has BOD_u = 320 mg/L with k = 0.23 /day (base e). What is BOD₅ and the remaining oxygen demand at day 5?

Start with the thinking

• BOD₅ is the amount exerted, not what remains.
• Exponential decay uses base e with this k.

Step-by-step solution

1. Decay factor

   $$e^{-kt} = e^{-0.23(5)} = e^{-1.15} = 0.3166$$

2. Exerted demand

   $$BOD_{5} = BOD_u(1 - e^{-kt}) = 320(1 - 0.3166)$$

3. Evaluate

   $$BOD_{5} = 320(0.6834) = 219 mg/L$$

4. Remaining

   $$320 - 219 = 101 mg/L$$

Why the other options are there

• 101 mg/L (remaining reported as BOD₅)
• 320 mg/L (ultimate reported)

Reference: FE Reference Handbook — Environmental — BOD kinetics

A environmental engineering problem uses First-order BOD decay. Given ultimate BOD (L0) = 300.0 mg/L; rate constant (k) = 0.1300 1/day; time (t) = 9.5000 day, determine the remaining BOD (Lt) in mg/L.

Start with the thinking

• The governing relation printed in this handbook section is First-order BOD decay.
• Everything except Lt is given, so isolate Lt 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.
• Environmental Engineering items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$L_t = L_0 e^{-k t}$$

2. Step 2 — Rearrange the relation so that Lt stands alone on the left-hand side.

3. Step 3

   $$List the givens: ultimate BOD (L_{0}) = 300.0 mg/L, rate constant (k) = 0.1300 1/day, time (t) = 9.5000 day$$

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$Lt = 87.2504\ \text{mg/L}$$

6. Step 6 — Check: returning Lt = 87.2504 mg/L to

   $$L_t = L_0 e^{-k t}$$

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

Why the other options are there

• 174.5 — kept a factor of two that cancels in the correct rearrangement.
• 43.6252 — dropped that same factor in the other direction.
• 95.9755 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → BOD Exertion

A environmental engineering problem uses First-order BOD decay. Given rate constant (k) = 0.1100 1/day; time (t) = 6.0000 day; remaining BOD (Lt) = 272.0 mg/L, determine the ultimate BOD (L0) in mg/L.

Start with the thinking

• The governing relation printed in this handbook section is First-order BOD decay.
• Everything except L0 is given, so isolate L0 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.
• Environmental Engineering items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$L_t = L_0 e^{-k t}$$

2. Step 2 — Rearrange the relation so that L0 stands alone on the left-hand side.

3. Step 3

   $$List the givens: rate constant (k) = 0.1100 1/day, time (t) = 6.0000 day, remaining BOD (Lt) = 272.0 mg/L$$

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$L_{0} = 526.3\ \text{mg/L}$$

6. Step 6 — Check: returning L0 = 526.3 mg/L to

   $$L_t = L_0 e^{-k t}$$

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

Why the other options are there

• 1,053 — kept a factor of two that cancels in the correct rearrangement.
• 263.1 — dropped that same factor in the other direction.
• 578.9 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → BOD Exertion

A environmental engineering problem uses First-order BOD decay. Given ultimate BOD (L0) = 295.0 mg/L; time (t) = 10.0000 day; remaining BOD (Lt) = 158.5 mg/L, determine the rate constant (k) in 1/day.

Start with the thinking

• The governing relation printed in this handbook section is First-order BOD decay.
• Everything except k is given, so isolate k 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.
• Environmental Engineering items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$L_t = L_0 e^{-k t}$$

2. Step 2 — Rearrange the relation so that k stands alone on the left-hand side.

3. Step 3

   $$List the givens: ultimate BOD (L_{0}) = 295.0 mg/L, time (t) = 10.0000 day, remaining BOD (Lt) = 158.5 mg/L$$

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$k = 0.0621\ \text{1/day}$$

6. Step 6 — Check: returning k = 0.0621 1/day to

   $$L_t = L_0 e^{-k t}$$

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

Why the other options are there

• 0.1242 — kept a factor of two that cancels in the correct rearrangement.
• 0.0311 — dropped that same factor in the other direction.
• 0.0683 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → BOD Exertion

A environmental engineering problem uses First-order BOD decay. Given ultimate BOD (L0) = 155.0 mg/L; rate constant (k) = 0.1400 1/day; time (t) = 4.5000 day, determine the remaining BOD (Lt) in mg/L.

Start with the thinking

• The governing relation printed in this handbook section is First-order BOD decay.
• Everything except Lt is given, so isolate Lt 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.
• Environmental Engineering items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$L_t = L_0 e^{-k t}$$

2. Step 2 — Rearrange the relation so that Lt stands alone on the left-hand side.

3. Step 3

   $$List the givens: ultimate BOD (L_{0}) = 155.0 mg/L, rate constant (k) = 0.1400 1/day, time (t) = 4.5000 day$$

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$Lt = 82.5517\ \text{mg/L}$$

6. Step 6 — Check: returning Lt = 82.5517 mg/L to

   $$L_t = L_0 e^{-k t}$$

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

Why the other options are there

• 165.1 — kept a factor of two that cancels in the correct rearrangement.
• 41.2759 — dropped that same factor in the other direction.
• 90.8069 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → BOD Exertion

A environmental engineering problem uses First-order BOD decay. Given rate constant (k) = 0.2000 1/day; time (t) = 5.5000 day; remaining BOD (Lt) = 40.7000 mg/L, determine the ultimate BOD (L0) in mg/L.

Start with the thinking

• The governing relation printed in this handbook section is First-order BOD decay.
• Everything except L0 is given, so isolate L0 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.
• Environmental Engineering items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$L_t = L_0 e^{-k t}$$

2. Step 2 — Rearrange the relation so that L0 stands alone on the left-hand side.

3. Step 3

   $$List the givens: rate constant (k) = 0.2000 1/day, time (t) = 5.5000 day, remaining BOD (Lt) = 40.7000 mg/L$$

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$L_{0} = 122.3\ \text{mg/L}$$

6. Step 6 — Check: returning L0 = 122.3 mg/L to

   $$L_t = L_0 e^{-k t}$$

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

Why the other options are there

• 244.5 — kept a factor of two that cancels in the correct rearrangement.
• 61.1348 — dropped that same factor in the other direction.
• 134.5 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → BOD Exertion

A environmental engineering problem uses First-order BOD decay. Given ultimate BOD (L0) = 395.0 mg/L; time (t) = 4.5000 day; remaining BOD (Lt) = 359.9 mg/L, determine the rate constant (k) in 1/day.

Start with the thinking

• The governing relation printed in this handbook section is First-order BOD decay.
• Everything except k is given, so isolate k 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.
• Environmental Engineering items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$L_t = L_0 e^{-k t}$$

2. Step 2 — Rearrange the relation so that k stands alone on the left-hand side.

3. Step 3

   $$List the givens: ultimate BOD (L_{0}) = 395.0 mg/L, time (t) = 4.5000 day, remaining BOD (Lt) = 359.9 mg/L$$

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$k = 0.0207\ \text{1/day}$$

6. Step 6 — Check: returning k = 0.0207 1/day to

   $$L_t = L_0 e^{-k t}$$

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

Why the other options are there

• 0.0414 — kept a factor of two that cancels in the correct rearrangement.
• 0.0103 — dropped that same factor in the other direction.
• 0.0227 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → BOD Exertion

A environmental engineering problem uses First-order BOD decay. Given ultimate BOD (L0) = 320.0 mg/L; rate constant (k) = 0.2200 1/day; time (t) = 2.5000 day, determine the remaining BOD (Lt) in mg/L.

Start with the thinking

• The governing relation printed in this handbook section is First-order BOD decay.
• Everything except Lt is given, so isolate Lt 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.
• Environmental Engineering items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$L_t = L_0 e^{-k t}$$

2. Step 2 — Rearrange the relation so that Lt stands alone on the left-hand side.

3. Step 3

   $$List the givens: ultimate BOD (L_{0}) = 320.0 mg/L, rate constant (k) = 0.2200 1/day, time (t) = 2.5000 day$$

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$Lt = 184.6\ \text{mg/L}$$

6. Step 6 — Check: returning Lt = 184.6 mg/L to

   $$L_t = L_0 e^{-k t}$$

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

Why the other options are there

• 369.2 — kept a factor of two that cancels in the correct rearrangement.
• 92.3120 — dropped that same factor in the other direction.
• 203.1 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → BOD Exertion

A environmental engineering problem uses First-order BOD decay. Given rate constant (k) = 0.1300 1/day; time (t) = 3.0000 day; remaining BOD (Lt) = 134.8 mg/L, determine the ultimate BOD (L0) in mg/L.

Start with the thinking

• The governing relation printed in this handbook section is First-order BOD decay.
• Everything except L0 is given, so isolate L0 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.
• Environmental Engineering items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$L_t = L_0 e^{-k t}$$

2. Step 2 — Rearrange the relation so that L0 stands alone on the left-hand side.

3. Step 3

   $$List the givens: rate constant (k) = 0.1300 1/day, time (t) = 3.0000 day, remaining BOD (Lt) = 134.8 mg/L$$

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$L_{0} = 199.1\ \text{mg/L}$$

6. Step 6 — Check: returning L0 = 199.1 mg/L to

   $$L_t = L_0 e^{-k t}$$

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

Why the other options are there

• 398.2 — kept a factor of two that cancels in the correct rearrangement.
• 99.5485 — dropped that same factor in the other direction.
• 219.0 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → BOD Exertion

A environmental engineering problem uses First-order BOD decay. Given ultimate BOD (L0) = 235.0 mg/L; time (t) = 2.5000 day; remaining BOD (Lt) = 384.7 mg/L, determine the rate constant (k) in 1/day.

Start with the thinking

• The governing relation printed in this handbook section is First-order BOD decay.
• Everything except k is given, so isolate k 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.
• Environmental Engineering items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$L_t = L_0 e^{-k t}$$

2. Step 2 — Rearrange the relation so that k stands alone on the left-hand side.

3. Step 3

   $$List the givens: ultimate BOD (L_{0}) = 235.0 mg/L, time (t) = 2.5000 day, remaining BOD (Lt) = 384.7 mg/L$$

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$k = -0.1972\ \text{1/day}$$

6. Step 6 — Check: returning k = -0.1972 1/day to

   $$L_t = L_0 e^{-k t}$$

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

Why the other options are there

• -0.3943 — kept a factor of two that cancels in the correct rearrangement.
• -0.0986 — dropped that same factor in the other direction.
• -0.2169 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Environmental Engineering → BOD Exertion