Freundlich isotherm for activated carbon adsorption of an organic compound Given Freundlich capacity coefficient (K_f) = 92.5000; equilibrium concentration (C_e) = 10.2000 mg/L; Freundlich intensity constant (n) = 4.3500, determine the adsorption capacity (x/m) (x_m) in mg/g.

Start with the thinking

• The governing relation printed in this handbook section is Freundlich Isotherm.
• Everything except x_m is given, so isolate x_m 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 Freundlich isotherm relates adsorption capacity to equilibrium concentration for activated carbon adsorption in a linearized form.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\dfrac{x}{m} = K_f C_e^{1/n}$$

2. Step 2 — Rearrange symbolically for x_m:

   $$x_{m} = K_f C_e^{1/n}$$

3. Step 3 — List the givens: Freundlich capacity coefficient (K_f) = 92.5000, equilibrium concentration (C_e) = 10.2000 mg/L, Freundlich intensity constant (n) = 4.3500.

4. Step 4 — Substitute the given values:

   $$x_{m} = K_f C_e^{1/4.3500}$$

5. Step 5 — Evaluate:

   $$x_{m} = 157.8\ \text{mg/g}$$

6. Step 6 — Check: returning x_m = 157.8 mg/g to

   $$\dfrac{x}{m} = K_f C_e^{1/n}$$

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

Why the other options are there

• 315.5 — kept a factor of two that cancels in the correct rearrangement.
• 78.8813 — dropped that same factor in the other direction.
• 173.5 — rounded an intermediate value before the final step.

Reference: FE Handbook — Freundlich Isotherm

Freundlich isotherm linearized form fit to laboratory adsorption data Given equilibrium concentration (C_e) = 21.2000 mg/L; Freundlich intensity constant (n) = 2.1500; adsorption capacity (x/m) (x_m) = 72.6000 mg/g, determine the Freundlich capacity coefficient (K_f).

Start with the thinking

• The governing relation printed in this handbook section is Freundlich Isotherm.
• Everything except K_f is given, so isolate K_f 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 Freundlich isotherm relates adsorption capacity to equilibrium concentration for activated carbon adsorption in a linearized form.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\dfrac{x}{m} = K_f C_e^{1/n}$$

2. Step 2 — Rearrange symbolically for K_f:

   $$K_{f} = \dfrac{x/m}{C_e^{1/n}}$$

3. Step 3 — List the givens: equilibrium concentration (C_e) = 21.2000 mg/L, Freundlich intensity constant (n) = 2.1500, adsorption capacity (x/m) (x_m) = 72.6000 mg/g.

4. Step 4 — Substitute the given values:

   $$K_{f} = \dfrac{x/m}{C_e^{1/2.1500}}$$

5. Step 5 — Evaluate:

   $$K_{f} = 17.5403$$

6. Step 6 — Check: returning K_f = 17.5403 to

   $$\dfrac{x}{m} = K_f C_e^{1/n}$$

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

Why the other options are there

• 35.0805 — kept a factor of two that cancels in the correct rearrangement.
• 8.7701 — dropped that same factor in the other direction.
• 19.2943 — rounded an intermediate value before the final step.

Reference: FE Handbook — Freundlich Isotherm

Freundlich isotherm used to design a granular activated carbon column Given Freundlich capacity coefficient (K_f) = 14.0000; Freundlich intensity constant (n) = 4.8500; adsorption capacity (x/m) (x_m) = 24.1000 mg/g, determine the equilibrium concentration (C_e) in mg/L.

Start with the thinking

• The governing relation printed in this handbook section is Freundlich Isotherm.
• Everything except C_e is given, so isolate C_e 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 Freundlich isotherm relates adsorption capacity to equilibrium concentration for activated carbon adsorption in a linearized form.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\dfrac{x}{m} = K_f C_e^{1/n}$$

2. Step 2 — Rearrange symbolically for C_e:

   $$C_{e} = \left(\dfrac{x/m}{K_f}\right)^{n}$$

3. Step 3 — List the givens: Freundlich capacity coefficient (K_f) = 14.0000, Freundlich intensity constant (n) = 4.8500, adsorption capacity (x/m) (x_m) = 24.1000 mg/g.

4. Step 4 — Substitute the given values:

   $$C_{e} = \left(\dfrac{x/m}{K_f}\right)^{4.8500}$$

5. Step 5 — Evaluate:

   $$C_{e} = 13.9335\ \text{mg/L}$$

6. Step 6 — Check: returning C_e = 13.9335 mg/L to

   $$\dfrac{x}{m} = K_f C_e^{1/n}$$

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

Why the other options are there

• 27.8671 — kept a factor of two that cancels in the correct rearrangement.
• 6.9668 — dropped that same factor in the other direction.
• 15.3269 — rounded an intermediate value before the final step.

Reference: FE Handbook — Freundlich Isotherm

Freundlich isotherm for activated carbon adsorption of an organic compound Given Freundlich capacity coefficient (K_f) = 9.0000; equilibrium concentration (C_e) = 29.5000 mg/L; Freundlich intensity constant (n) = 2.9500, determine the adsorption capacity (x/m) (x_m) in mg/g.

Start with the thinking

• The governing relation printed in this handbook section is Freundlich Isotherm.
• Everything except x_m is given, so isolate x_m 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 Freundlich isotherm relates adsorption capacity to equilibrium concentration for activated carbon adsorption in a linearized form.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\dfrac{x}{m} = K_f C_e^{1/n}$$

2. Step 2 — Rearrange symbolically for x_m:

   $$x_{m} = K_f C_e^{1/n}$$

3. Step 3 — List the givens: Freundlich capacity coefficient (K_f) = 9.0000, equilibrium concentration (C_e) = 29.5000 mg/L, Freundlich intensity constant (n) = 2.9500.

4. Step 4 — Substitute the given values:

   $$x_{m} = K_f C_e^{1/2.9500}$$

5. Step 5 — Evaluate:

   $$x_{m} = 28.3457\ \text{mg/g}$$

6. Step 6 — Check: returning x_m = 28.3457 mg/g to

   $$\dfrac{x}{m} = K_f C_e^{1/n}$$

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

Why the other options are there

• 56.6914 — kept a factor of two that cancels in the correct rearrangement.
• 14.1729 — dropped that same factor in the other direction.
• 31.1803 — rounded an intermediate value before the final step.

Reference: FE Handbook — Freundlich Isotherm

Freundlich isotherm linearized form fit to laboratory adsorption data Given equilibrium concentration (C_e) = 32.5000 mg/L; Freundlich intensity constant (n) = 1.2000; adsorption capacity (x/m) (x_m) = 40.9000 mg/g, determine the Freundlich capacity coefficient (K_f).

Start with the thinking

• The governing relation printed in this handbook section is Freundlich Isotherm.
• Everything except K_f is given, so isolate K_f 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 Freundlich isotherm relates adsorption capacity to equilibrium concentration for activated carbon adsorption in a linearized form.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\dfrac{x}{m} = K_f C_e^{1/n}$$

2. Step 2 — Rearrange symbolically for K_f:

   $$K_{f} = \dfrac{x/m}{C_e^{1/n}}$$

3. Step 3 — List the givens: equilibrium concentration (C_e) = 32.5000 mg/L, Freundlich intensity constant (n) = 1.2000, adsorption capacity (x/m) (x_m) = 40.9000 mg/g.

4. Step 4 — Substitute the given values:

   $$K_{f} = \dfrac{x/m}{C_e^{1/1.2000}}$$

5. Step 5 — Evaluate:

   $$K_{f} = 2.2481$$

6. Step 6 — Check: returning K_f = 2.2481 to

   $$\dfrac{x}{m} = K_f C_e^{1/n}$$

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

Why the other options are there

• 4.4963 — kept a factor of two that cancels in the correct rearrangement.
• 1.1241 — dropped that same factor in the other direction.
• 2.4729 — rounded an intermediate value before the final step.

Reference: FE Handbook — Freundlich Isotherm

Freundlich isotherm used to design a granular activated carbon column Given Freundlich capacity coefficient (K_f) = 73.0000; Freundlich intensity constant (n) = 1.2000; adsorption capacity (x/m) (x_m) = 62.9000 mg/g, determine the equilibrium concentration (C_e) in mg/L.

Start with the thinking

• The governing relation printed in this handbook section is Freundlich Isotherm.
• Everything except C_e is given, so isolate C_e 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 Freundlich isotherm relates adsorption capacity to equilibrium concentration for activated carbon adsorption in a linearized form.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\dfrac{x}{m} = K_f C_e^{1/n}$$

2. Step 2 — Rearrange symbolically for C_e:

   $$C_{e} = \left(\dfrac{x/m}{K_f}\right)^{n}$$

3. Step 3 — List the givens: Freundlich capacity coefficient (K_f) = 73.0000, Freundlich intensity constant (n) = 1.2000, adsorption capacity (x/m) (x_m) = 62.9000 mg/g.

4. Step 4 — Substitute the given values:

   $$C_{e} = \left(\dfrac{x/m}{K_f}\right)^{1.2000}$$

5. Step 5 — Evaluate:

   $$C_{e} = 0.8364\ \text{mg/L}$$

6. Step 6 — Check: returning C_e = 0.8364 mg/L to

   $$\dfrac{x}{m} = K_f C_e^{1/n}$$

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

Why the other options are there

• 1.6727 — kept a factor of two that cancels in the correct rearrangement.
• 0.4182 — dropped that same factor in the other direction.
• 0.9200 — rounded an intermediate value before the final step.

Reference: FE Handbook — Freundlich Isotherm

Freundlich isotherm for activated carbon adsorption of an organic compound Given Freundlich capacity coefficient (K_f) = 96.0000; equilibrium concentration (C_e) = 19.9000 mg/L; Freundlich intensity constant (n) = 3.9500, determine the adsorption capacity (x/m) (x_m) in mg/g.

Start with the thinking

• The governing relation printed in this handbook section is Freundlich Isotherm.
• Everything except x_m is given, so isolate x_m 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 Freundlich isotherm relates adsorption capacity to equilibrium concentration for activated carbon adsorption in a linearized form.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\dfrac{x}{m} = K_f C_e^{1/n}$$

2. Step 2 — Rearrange symbolically for x_m:

   $$x_{m} = K_f C_e^{1/n}$$

3. Step 3 — List the givens: Freundlich capacity coefficient (K_f) = 96.0000, equilibrium concentration (C_e) = 19.9000 mg/L, Freundlich intensity constant (n) = 3.9500.

4. Step 4 — Substitute the given values:

   $$x_{m} = K_f C_e^{1/3.9500}$$

5. Step 5 — Evaluate:

   $$x_{m} = 204.7\ \text{mg/g}$$

6. Step 6 — Check: returning x_m = 204.7 mg/g to

   $$\dfrac{x}{m} = K_f C_e^{1/n}$$

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

Why the other options are there

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

Reference: FE Handbook — Freundlich Isotherm

Freundlich isotherm linearized form fit to laboratory adsorption data Given equilibrium concentration (C_e) = 21.4000 mg/L; Freundlich intensity constant (n) = 4.0500; adsorption capacity (x/m) (x_m) = 159.0 mg/g, determine the Freundlich capacity coefficient (K_f).

Start with the thinking

• The governing relation printed in this handbook section is Freundlich Isotherm.
• Everything except K_f is given, so isolate K_f 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 Freundlich isotherm relates adsorption capacity to equilibrium concentration for activated carbon adsorption in a linearized form.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\dfrac{x}{m} = K_f C_e^{1/n}$$

2. Step 2 — Rearrange symbolically for K_f:

   $$K_{f} = \dfrac{x/m}{C_e^{1/n}}$$

3. Step 3 — List the givens: equilibrium concentration (C_e) = 21.4000 mg/L, Freundlich intensity constant (n) = 4.0500, adsorption capacity (x/m) (x_m) = 159.0 mg/g.

4. Step 4 — Substitute the given values:

   $$K_{f} = \dfrac{x/m}{C_e^{1/4.0500}}$$

5. Step 5 — Evaluate:

   $$K_{f} = 74.6277$$

6. Step 6 — Check: returning K_f = 74.6277 to

   $$\dfrac{x}{m} = K_f C_e^{1/n}$$

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

Why the other options are there

• 149.3 — kept a factor of two that cancels in the correct rearrangement.
• 37.3138 — dropped that same factor in the other direction.
• 82.0904 — rounded an intermediate value before the final step.

Reference: FE Handbook — Freundlich Isotherm

Freundlich isotherm used to design a granular activated carbon column Given Freundlich capacity coefficient (K_f) = 58.5000; Freundlich intensity constant (n) = 4.7000; adsorption capacity (x/m) (x_m) = 153.1 mg/g, determine the equilibrium concentration (C_e) in mg/L.

Start with the thinking

• The governing relation printed in this handbook section is Freundlich Isotherm.
• Everything except C_e is given, so isolate C_e 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 Freundlich isotherm relates adsorption capacity to equilibrium concentration for activated carbon adsorption in a linearized form.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\dfrac{x}{m} = K_f C_e^{1/n}$$

2. Step 2 — Rearrange symbolically for C_e:

   $$C_{e} = \left(\dfrac{x/m}{K_f}\right)^{n}$$

3. Step 3 — List the givens: Freundlich capacity coefficient (K_f) = 58.5000, Freundlich intensity constant (n) = 4.7000, adsorption capacity (x/m) (x_m) = 153.1 mg/g.

4. Step 4 — Substitute the given values:

   $$C_{e} = \left(\dfrac{x/m}{K_f}\right)^{4.7000}$$

5. Step 5 — Evaluate:

   $$C_{e} = 91.9922\ \text{mg/L}$$

6. Step 6 — Check: returning C_e = 91.9922 mg/L to

   $$\dfrac{x}{m} = K_f C_e^{1/n}$$

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

Why the other options are there

• 184.0 — kept a factor of two that cancels in the correct rearrangement.
• 45.9961 — dropped that same factor in the other direction.
• 101.2 — rounded an intermediate value before the final step.

Reference: FE Handbook — Freundlich Isotherm

Freundlich isotherm for activated carbon adsorption of an organic compound Given Freundlich capacity coefficient (K_f) = 61.5000; equilibrium concentration (C_e) = 8.4000 mg/L; Freundlich intensity constant (n) = 4.3000, determine the adsorption capacity (x/m) (x_m) in mg/g.

Start with the thinking

• The governing relation printed in this handbook section is Freundlich Isotherm.
• Everything except x_m is given, so isolate x_m 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 Freundlich isotherm relates adsorption capacity to equilibrium concentration for activated carbon adsorption in a linearized form.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\dfrac{x}{m} = K_f C_e^{1/n}$$

2. Step 2 — Rearrange symbolically for x_m:

   $$x_{m} = K_f C_e^{1/n}$$

3. Step 3 — List the givens: Freundlich capacity coefficient (K_f) = 61.5000, equilibrium concentration (C_e) = 8.4000 mg/L, Freundlich intensity constant (n) = 4.3000.

4. Step 4 — Substitute the given values:

   $$x_{m} = K_f C_e^{1/4.3000}$$

5. Step 5 — Evaluate:

   $$x_{m} = 100.9\ \text{mg/g}$$

6. Step 6 — Check: returning x_m = 100.9 mg/g to

   $$\dfrac{x}{m} = K_f C_e^{1/n}$$

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

Why the other options are there

• 201.8 — kept a factor of two that cancels in the correct rearrangement.
• 50.4422 — dropped that same factor in the other direction.
• 111.0 — rounded an intermediate value before the final step.

Reference: FE Handbook — Freundlich Isotherm