kinetic temperature correction of a BOD rate constant using theta Given rate constant at T1 (k_T1) = 0.7000 1/day; temperature correction coefficient (theta) = 1.0910; temperature 2 (T2) = 31.0000 C; temperature 1 (T1) = 7.0000 C, determine the rate constant at T2 (k_T2) in 1/day.

Start with the thinking

• The governing relation printed in this handbook section is Kinetic Temperature Corrections (Arrhenius theta).
• Everything except k_T2 is given, so isolate k_T2 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.
• Kinetic temperature corrections use the Arrhenius theta relationship to adjust reaction rate constants between two temperatures.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$k_{T_2} = k_{T_1} \theta^{(T_2 - T_1)}$$

2. Step 2 — Rearrange symbolically for k_T2:

   $$k_{T2} = k_{T_1}\theta^{(T_2-T_1)}$$

3. Step 3 — List the givens: rate constant at T1 (k_T1) = 0.7000 1/day, temperature correction coefficient (theta) = 1.0910, temperature 2 (T2) = 31.0000 C, temperature 1 (T1) = 7.0000 C.

4. Step 4 — Substitute the given values:

   $$k_{T2} = k_{T_1}1.0910^{(T_2-T_1)}$$

5. Step 5 — Evaluate:

   $$k_{T2} = 5.6610\ \text{1/day}$$

6. Step 6 — Check: returning k_T2 = 5.6610 1/day to

   $$k_{T_2} = k_{T_1} \theta^{(T_2 - T_1)}$$

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

Why the other options are there

• 11.3220 — kept a factor of two that cancels in the correct rearrangement.
• 2.8305 — dropped that same factor in the other direction.
• 6.2271 — rounded an intermediate value before the final step.

Reference: FE Handbook — Kinetic Temperature Corrections

Arrhenius-type temperature correction for a biological treatment rate constant Given temperature correction coefficient (theta) = 1.0530; temperature 2 (T2) = 13.5000 C; temperature 1 (T1) = 12.0000 C; rate constant at T2 (k_T2) = 4.3800 1/day, determine the rate constant at T1 (k_T1) in 1/day.

Start with the thinking

• The governing relation printed in this handbook section is Kinetic Temperature Corrections (Arrhenius theta).
• Everything except k_T1 is given, so isolate k_T1 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.
• Kinetic temperature corrections use the Arrhenius theta relationship to adjust reaction rate constants between two temperatures.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$k_{T_2} = k_{T_1} \theta^{(T_2 - T_1)}$$

2. Step 2 — Rearrange symbolically for k_T1:

   $$k_{T1} = \dfrac{k_{T_2}}{\theta^{(T_2-T_1)}}$$

3. Step 3 — List the givens: temperature correction coefficient (theta) = 1.0530, temperature 2 (T2) = 13.5000 C, temperature 1 (T1) = 12.0000 C, rate constant at T2 (k_T2) = 4.3800 1/day.

4. Step 4 — Substitute the given values:

   $$k_{T1} = \dfrac{k_{T_2}}{1.0530^{(T_2-T_1)}}$$

5. Step 5 — Evaluate:

   $$k_{T1} = 4.0535\ \text{1/day}$$

6. Step 6 — Check: returning k_T1 = 4.0535 1/day to

   $$k_{T_2} = k_{T_1} \theta^{(T_2 - T_1)}$$

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

Why the other options are there

• 8.1070 — kept a factor of two that cancels in the correct rearrangement.
• 2.0268 — dropped that same factor in the other direction.
• 4.4589 — rounded an intermediate value before the final step.

Reference: FE Handbook — Kinetic Temperature Corrections

temperature correction of reaction rate constant from summer to winter conditions Given rate constant at T1 (k_T1) = 1.6800 1/day; temperature 2 (T2) = 26.0000 C; temperature 1 (T1) = 20.0000 C; rate constant at T2 (k_T2) = 0.8270 1/day, determine the temperature correction coefficient (theta).

Start with the thinking

• The governing relation printed in this handbook section is Kinetic Temperature Corrections (Arrhenius theta).
• Everything except theta is given, so isolate theta 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.
• Kinetic temperature corrections use the Arrhenius theta relationship to adjust reaction rate constants between two temperatures.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$k_{T_2} = k_{T_1} \theta^{(T_2 - T_1)}$$

2. Step 2 — Rearrange symbolically for theta:

   $$\theta = \left(\dfrac{k_{T_2}}{k_{T_1}}\right)^{1/(T_2-T_1)}$$

3. Step 3 — List the givens: rate constant at T1 (k_T1) = 1.6800 1/day, temperature 2 (T2) = 26.0000 C, temperature 1 (T1) = 20.0000 C, rate constant at T2 (k_T2) = 0.8270 1/day.

4. Step 4 — Substitute the given values:

   $$\theta = \left(\dfrac{k_{T_2}}{k_{T_1}}\right)^{1/(T_2-T_1)}$$

5. Step 5 — Evaluate:

   $$\theta = 0.8886$$

6. Step 6 — Check: returning theta = 0.8886 to

   $$k_{T_2} = k_{T_1} \theta^{(T_2 - T_1)}$$

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

Why the other options are there

• 1.7772 — kept a factor of two that cancels in the correct rearrangement.
• 0.4443 — dropped that same factor in the other direction.
• 0.9774 — rounded an intermediate value before the final step.

Reference: FE Handbook — Kinetic Temperature Corrections

kinetic temperature correction of a BOD rate constant using theta Given rate constant at T1 (k_T1) = 1.4900 1/day; temperature correction coefficient (theta) = 1.0970; temperature 2 (T2) = 27.5000 C; temperature 1 (T1) = 13.0000 C, determine the rate constant at T2 (k_T2) in 1/day.

Start with the thinking

• The governing relation printed in this handbook section is Kinetic Temperature Corrections (Arrhenius theta).
• Everything except k_T2 is given, so isolate k_T2 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.
• Kinetic temperature corrections use the Arrhenius theta relationship to adjust reaction rate constants between two temperatures.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$k_{T_2} = k_{T_1} \theta^{(T_2 - T_1)}$$

2. Step 2 — Rearrange symbolically for k_T2:

   $$k_{T2} = k_{T_1}\theta^{(T_2-T_1)}$$

3. Step 3 — List the givens: rate constant at T1 (k_T1) = 1.4900 1/day, temperature correction coefficient (theta) = 1.0970, temperature 2 (T2) = 27.5000 C, temperature 1 (T1) = 13.0000 C.

4. Step 4 — Substitute the given values:

   $$k_{T2} = k_{T_1}1.0970^{(T_2-T_1)}$$

5. Step 5 — Evaluate:

   $$k_{T2} = 5.7040\ \text{1/day}$$

6. Step 6 — Check: returning k_T2 = 5.7040 1/day to

   $$k_{T_2} = k_{T_1} \theta^{(T_2 - T_1)}$$

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

Why the other options are there

• 11.4081 — kept a factor of two that cancels in the correct rearrangement.
• 2.8520 — dropped that same factor in the other direction.
• 6.2744 — rounded an intermediate value before the final step.

Reference: FE Handbook — Kinetic Temperature Corrections

Arrhenius-type temperature correction for a biological treatment rate constant Given temperature correction coefficient (theta) = 1.1450; temperature 2 (T2) = 29.0000 C; temperature 1 (T1) = 13.0000 C; rate constant at T2 (k_T2) = 4.8710 1/day, determine the rate constant at T1 (k_T1) in 1/day.

Start with the thinking

• The governing relation printed in this handbook section is Kinetic Temperature Corrections (Arrhenius theta).
• Everything except k_T1 is given, so isolate k_T1 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.
• Kinetic temperature corrections use the Arrhenius theta relationship to adjust reaction rate constants between two temperatures.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$k_{T_2} = k_{T_1} \theta^{(T_2 - T_1)}$$

2. Step 2 — Rearrange symbolically for k_T1:

   $$k_{T1} = \dfrac{k_{T_2}}{\theta^{(T_2-T_1)}}$$

3. Step 3 — List the givens: temperature correction coefficient (theta) = 1.1450, temperature 2 (T2) = 29.0000 C, temperature 1 (T1) = 13.0000 C, rate constant at T2 (k_T2) = 4.8710 1/day.

4. Step 4 — Substitute the given values:

   $$k_{T1} = \dfrac{k_{T_2}}{1.1450^{(T_2-T_1)}}$$

5. Step 5 — Evaluate:

   $$k_{T1} = 0.5581\ \text{1/day}$$

6. Step 6 — Check: returning k_T1 = 0.5581 1/day to

   $$k_{T_2} = k_{T_1} \theta^{(T_2 - T_1)}$$

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

Why the other options are there

• 1.1162 — kept a factor of two that cancels in the correct rearrangement.
• 0.2791 — dropped that same factor in the other direction.
• 0.6139 — rounded an intermediate value before the final step.

Reference: FE Handbook — Kinetic Temperature Corrections

temperature correction of reaction rate constant from summer to winter conditions Given rate constant at T1 (k_T1) = 1.9300 1/day; temperature 2 (T2) = 22.0000 C; temperature 1 (T1) = 29.5000 C; rate constant at T2 (k_T2) = 0.7010 1/day, determine the temperature correction coefficient (theta).

Start with the thinking

• The governing relation printed in this handbook section is Kinetic Temperature Corrections (Arrhenius theta).
• Everything except theta is given, so isolate theta 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.
• Kinetic temperature corrections use the Arrhenius theta relationship to adjust reaction rate constants between two temperatures.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$k_{T_2} = k_{T_1} \theta^{(T_2 - T_1)}$$

2. Step 2 — Rearrange symbolically for theta:

   $$\theta = \left(\dfrac{k_{T_2}}{k_{T_1}}\right)^{1/(T_2-T_1)}$$

3. Step 3 — List the givens: rate constant at T1 (k_T1) = 1.9300 1/day, temperature 2 (T2) = 22.0000 C, temperature 1 (T1) = 29.5000 C, rate constant at T2 (k_T2) = 0.7010 1/day.

4. Step 4 — Substitute the given values:

   $$\theta = \left(\dfrac{k_{T_2}}{k_{T_1}}\right)^{1/(T_2-T_1)}$$

5. Step 5 — Evaluate:

   $$\theta = 1.1446$$

6. Step 6 — Check: returning theta = 1.1446 to

   $$k_{T_2} = k_{T_1} \theta^{(T_2 - T_1)}$$

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

Why the other options are there

• 2.2892 — kept a factor of two that cancels in the correct rearrangement.
• 0.5723 — dropped that same factor in the other direction.
• 1.2590 — rounded an intermediate value before the final step.

Reference: FE Handbook — Kinetic Temperature Corrections

kinetic temperature correction of a BOD rate constant using theta Given rate constant at T1 (k_T1) = 0.0900 1/day; temperature correction coefficient (theta) = 1.0720; temperature 2 (T2) = 27.0000 C; temperature 1 (T1) = 11.5000 C, determine the rate constant at T2 (k_T2) in 1/day.

Start with the thinking

• The governing relation printed in this handbook section is Kinetic Temperature Corrections (Arrhenius theta).
• Everything except k_T2 is given, so isolate k_T2 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.
• Kinetic temperature corrections use the Arrhenius theta relationship to adjust reaction rate constants between two temperatures.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$k_{T_2} = k_{T_1} \theta^{(T_2 - T_1)}$$

2. Step 2 — Rearrange symbolically for k_T2:

   $$k_{T2} = k_{T_1}\theta^{(T_2-T_1)}$$

3. Step 3 — List the givens: rate constant at T1 (k_T1) = 0.0900 1/day, temperature correction coefficient (theta) = 1.0720, temperature 2 (T2) = 27.0000 C, temperature 1 (T1) = 11.5000 C.

4. Step 4 — Substitute the given values:

   $$k_{T2} = k_{T_1}1.0720^{(T_2-T_1)}$$

5. Step 5 — Evaluate:

   $$k_{T2} = 0.2644\ \text{1/day}$$

6. Step 6 — Check: returning k_T2 = 0.2644 1/day to

   $$k_{T_2} = k_{T_1} \theta^{(T_2 - T_1)}$$

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

Why the other options are there

• 0.5288 — kept a factor of two that cancels in the correct rearrangement.
• 0.1322 — dropped that same factor in the other direction.
• 0.2908 — rounded an intermediate value before the final step.

Reference: FE Handbook — Kinetic Temperature Corrections

Arrhenius-type temperature correction for a biological treatment rate constant Given temperature correction coefficient (theta) = 1.0760; temperature 2 (T2) = 16.5000 C; temperature 1 (T1) = 13.5000 C; rate constant at T2 (k_T2) = 2.8380 1/day, determine the rate constant at T1 (k_T1) in 1/day.

Start with the thinking

• The governing relation printed in this handbook section is Kinetic Temperature Corrections (Arrhenius theta).
• Everything except k_T1 is given, so isolate k_T1 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.
• Kinetic temperature corrections use the Arrhenius theta relationship to adjust reaction rate constants between two temperatures.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$k_{T_2} = k_{T_1} \theta^{(T_2 - T_1)}$$

2. Step 2 — Rearrange symbolically for k_T1:

   $$k_{T1} = \dfrac{k_{T_2}}{\theta^{(T_2-T_1)}}$$

3. Step 3 — List the givens: temperature correction coefficient (theta) = 1.0760, temperature 2 (T2) = 16.5000 C, temperature 1 (T1) = 13.5000 C, rate constant at T2 (k_T2) = 2.8380 1/day.

4. Step 4 — Substitute the given values:

   $$k_{T1} = \dfrac{k_{T_2}}{1.0760^{(T_2-T_1)}}$$

5. Step 5 — Evaluate:

   $$k_{T1} = 2.2781\ \text{1/day}$$

6. Step 6 — Check: returning k_T1 = 2.2781 1/day to

   $$k_{T_2} = k_{T_1} \theta^{(T_2 - T_1)}$$

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

Why the other options are there

• 4.5562 — kept a factor of two that cancels in the correct rearrangement.
• 1.1391 — dropped that same factor in the other direction.
• 2.5059 — rounded an intermediate value before the final step.

Reference: FE Handbook — Kinetic Temperature Corrections

temperature correction of reaction rate constant from summer to winter conditions Given rate constant at T1 (k_T1) = 1.6200 1/day; temperature 2 (T2) = 34.0000 C; temperature 1 (T1) = 5.5000 C; rate constant at T2 (k_T2) = 4.7870 1/day, determine the temperature correction coefficient (theta).

Start with the thinking

• The governing relation printed in this handbook section is Kinetic Temperature Corrections (Arrhenius theta).
• Everything except theta is given, so isolate theta 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.
• Kinetic temperature corrections use the Arrhenius theta relationship to adjust reaction rate constants between two temperatures.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$k_{T_2} = k_{T_1} \theta^{(T_2 - T_1)}$$

2. Step 2 — Rearrange symbolically for theta:

   $$\theta = \left(\dfrac{k_{T_2}}{k_{T_1}}\right)^{1/(T_2-T_1)}$$

3. Step 3 — List the givens: rate constant at T1 (k_T1) = 1.6200 1/day, temperature 2 (T2) = 34.0000 C, temperature 1 (T1) = 5.5000 C, rate constant at T2 (k_T2) = 4.7870 1/day.

4. Step 4 — Substitute the given values:

   $$\theta = \left(\dfrac{k_{T_2}}{k_{T_1}}\right)^{1/(T_2-T_1)}$$

5. Step 5 — Evaluate:

   $$\theta = 1.0387$$

6. Step 6 — Check: returning theta = 1.0387 to

   $$k_{T_2} = k_{T_1} \theta^{(T_2 - T_1)}$$

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

Why the other options are there

• 2.0775 — kept a factor of two that cancels in the correct rearrangement.
• 0.5194 — dropped that same factor in the other direction.
• 1.1426 — rounded an intermediate value before the final step.

Reference: FE Handbook — Kinetic Temperature Corrections

kinetic temperature correction of a BOD rate constant using theta Given rate constant at T1 (k_T1) = 1.5900 1/day; temperature correction coefficient (theta) = 1.0680; temperature 2 (T2) = 6.5000 C; temperature 1 (T1) = 24.0000 C, determine the rate constant at T2 (k_T2) in 1/day.

Start with the thinking

• The governing relation printed in this handbook section is Kinetic Temperature Corrections (Arrhenius theta).
• Everything except k_T2 is given, so isolate k_T2 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.
• Kinetic temperature corrections use the Arrhenius theta relationship to adjust reaction rate constants between two temperatures.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$k_{T_2} = k_{T_1} \theta^{(T_2 - T_1)}$$

2. Step 2 — Rearrange symbolically for k_T2:

   $$k_{T2} = k_{T_1}\theta^{(T_2-T_1)}$$

3. Step 3 — List the givens: rate constant at T1 (k_T1) = 1.5900 1/day, temperature correction coefficient (theta) = 1.0680, temperature 2 (T2) = 6.5000 C, temperature 1 (T1) = 24.0000 C.

4. Step 4 — Substitute the given values:

   $$k_{T2} = k_{T_1}1.0680^{(T_2-T_1)}$$

5. Step 5 — Evaluate:

   $$k_{T2} = 0.5028\ \text{1/day}$$

6. Step 6 — Check: returning k_T2 = 0.5028 1/day to

   $$k_{T_2} = k_{T_1} \theta^{(T_2 - T_1)}$$

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

Why the other options are there

• 1.0056 — kept a factor of two that cancels in the correct rearrangement.
• 0.2514 — dropped that same factor in the other direction.
• 0.5531 — rounded an intermediate value before the final step.

Reference: FE Handbook — Kinetic Temperature Corrections