Langmuir isotherm monolayer adsorption model for carbon treatment Given maximum adsorption capacity (a) = 12.0000 mg/g; Langmuir equilibrium constant (b) = 0.0540 L/mg; equilibrium concentration (C_e) = 44.3000 mg/L, determine the adsorption capacity (x/m) (x_m) in mg/g.

Start with the thinking

• The governing relation printed in this handbook section is Langmuir 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 Langmuir isotherm describes monolayer adsorption capacity and can be expressed in a linearized form for parameter fitting.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\dfrac{x}{m} = \dfrac{a b C_e}{1 + b C_e}$$

2. Step 2 — Rearrange symbolically for x_m:

   $$x_{m} = \dfrac{abC_e}{1+bC_e}$$

3. Step 3 — List the givens: maximum adsorption capacity (a) = 12.0000 mg/g, Langmuir equilibrium constant (b) = 0.0540 L/mg, equilibrium concentration (C_e) = 44.3000 mg/L.

4. Step 4 — Substitute the given values:

   $$x_{m} = \dfrac{abC_e}{1+bC_e}$$

5. Step 5 — Evaluate:

   $$x_{m} = 8.4625\ \text{mg/g}$$

6. Step 6 — Check: returning x_m = 8.4625 mg/g to

   $$\dfrac{x}{m} = \dfrac{a b C_e}{1 + b C_e}$$

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

Why the other options are there

• 16.9249 — kept a factor of two that cancels in the correct rearrangement.
• 4.2312 — dropped that same factor in the other direction.
• 9.3087 — rounded an intermediate value before the final step.

Reference: FE Handbook — Langmuir Isotherm

Langmuir isotherm linearized form used to determine adsorption constants Given maximum adsorption capacity (a) = 163.0 mg/g; Langmuir equilibrium constant (b) = 1.9480 L/mg; adsorption capacity (x/m) (x_m) = 132.2 mg/g, determine the equilibrium concentration (C_e) in mg/L.

Start with the thinking

• The governing relation printed in this handbook section is Langmuir 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 Langmuir isotherm describes monolayer adsorption capacity and can be expressed in a linearized form for parameter fitting.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\dfrac{x}{m} = \dfrac{a b C_e}{1 + b C_e}$$

2. Step 2 — Rearrange symbolically for C_e:

   $$C_{e} = \dfrac{x/m}{ab-(x/m)b}$$

3. Step 3 — List the givens: maximum adsorption capacity (a) = 163.0 mg/g, Langmuir equilibrium constant (b) = 1.9480 L/mg, adsorption capacity (x/m) (x_m) = 132.2 mg/g.

4. Step 4 — Substitute the given values:

   $$C_{e} = \dfrac{x/m}{a1.9480-(x/m)1.9480}$$

5. Step 5 — Evaluate:

   $$C_{e} = 2.2034\ \text{mg/L}$$

6. Step 6 — Check: returning C_e = 2.2034 mg/L to

   $$\dfrac{x}{m} = \dfrac{a b C_e}{1 + b C_e}$$

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

Why the other options are there

• 4.4068 — kept a factor of two that cancels in the correct rearrangement.
• 1.1017 — dropped that same factor in the other direction.
• 2.4237 — rounded an intermediate value before the final step.

Reference: FE Handbook — Langmuir Isotherm

Langmuir isotherm adsorption capacity for water treatment design Given Langmuir equilibrium constant (b) = 1.2160 L/mg; equilibrium concentration (C_e) = 4.9000 mg/L; adsorption capacity (x/m) (x_m) = 169.5 mg/g, determine the maximum adsorption capacity (a) in mg/g.

Start with the thinking

• The governing relation printed in this handbook section is Langmuir Isotherm.
• Everything except a is given, so isolate a 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 Langmuir isotherm describes monolayer adsorption capacity and can be expressed in a linearized form for parameter fitting.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\dfrac{x}{m} = \dfrac{a b C_e}{1 + b C_e}$$

2. Step 2 — Rearrange symbolically for a:

   $$a = \dfrac{(x/m)(1+bC_e)}{bC_e}$$

3. Step 3 — List the givens: Langmuir equilibrium constant (b) = 1.2160 L/mg, equilibrium concentration (C_e) = 4.9000 mg/L, adsorption capacity (x/m) (x_m) = 169.5 mg/g.

4. Step 4 — Substitute the given values:

   $$a = \dfrac{(x/m)(1+bC_e)}{bC_e}$$

5. Step 5 — Evaluate:

   $$a = 197.9\ \text{mg/g}$$

6. Step 6 — Check: returning a = 197.9 mg/g to

   $$\dfrac{x}{m} = \dfrac{a b C_e}{1 + b C_e}$$

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

Why the other options are there

• 395.9 — kept a factor of two that cancels in the correct rearrangement.
• 98.9736 — dropped that same factor in the other direction.
• 217.7 — rounded an intermediate value before the final step.

Reference: FE Handbook — Langmuir Isotherm

Langmuir isotherm monolayer adsorption model for carbon treatment Given maximum adsorption capacity (a) = 104.0 mg/g; Langmuir equilibrium constant (b) = 1.7520 L/mg; equilibrium concentration (C_e) = 22.9000 mg/L, determine the adsorption capacity (x/m) (x_m) in mg/g.

Start with the thinking

• The governing relation printed in this handbook section is Langmuir 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 Langmuir isotherm describes monolayer adsorption capacity and can be expressed in a linearized form for parameter fitting.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\dfrac{x}{m} = \dfrac{a b C_e}{1 + b C_e}$$

2. Step 2 — Rearrange symbolically for x_m:

   $$x_{m} = \dfrac{abC_e}{1+bC_e}$$

3. Step 3 — List the givens: maximum adsorption capacity (a) = 104.0 mg/g, Langmuir equilibrium constant (b) = 1.7520 L/mg, equilibrium concentration (C_e) = 22.9000 mg/L.

4. Step 4 — Substitute the given values:

   $$x_{m} = \dfrac{abC_e}{1+bC_e}$$

5. Step 5 — Evaluate:

   $$x_{m} = 101.5\ \text{mg/g}$$

6. Step 6 — Check: returning x_m = 101.5 mg/g to

   $$\dfrac{x}{m} = \dfrac{a b C_e}{1 + b C_e}$$

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

Why the other options are there

• 202.9 — kept a factor of two that cancels in the correct rearrangement.
• 50.7354 — dropped that same factor in the other direction.
• 111.6 — rounded an intermediate value before the final step.

Reference: FE Handbook — Langmuir Isotherm

Langmuir isotherm linearized form used to determine adsorption constants Given maximum adsorption capacity (a) = 159.0 mg/g; Langmuir equilibrium constant (b) = 0.5890 L/mg; adsorption capacity (x/m) (x_m) = 86.7000 mg/g, determine the equilibrium concentration (C_e) in mg/L.

Start with the thinking

• The governing relation printed in this handbook section is Langmuir 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 Langmuir isotherm describes monolayer adsorption capacity and can be expressed in a linearized form for parameter fitting.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\dfrac{x}{m} = \dfrac{a b C_e}{1 + b C_e}$$

2. Step 2 — Rearrange symbolically for C_e:

   $$C_{e} = \dfrac{x/m}{ab-(x/m)b}$$

3. Step 3 — List the givens: maximum adsorption capacity (a) = 159.0 mg/g, Langmuir equilibrium constant (b) = 0.5890 L/mg, adsorption capacity (x/m) (x_m) = 86.7000 mg/g.

4. Step 4 — Substitute the given values:

   $$C_{e} = \dfrac{x/m}{a0.5890-(x/m)0.5890}$$

5. Step 5 — Evaluate:

   $$C_{e} = 2.0359\ \text{mg/L}$$

6. Step 6 — Check: returning C_e = 2.0359 mg/L to

   $$\dfrac{x}{m} = \dfrac{a b C_e}{1 + b C_e}$$

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

Why the other options are there

• 4.0719 — kept a factor of two that cancels in the correct rearrangement.
• 1.0180 — dropped that same factor in the other direction.
• 2.2395 — rounded an intermediate value before the final step.

Reference: FE Handbook — Langmuir Isotherm

Langmuir isotherm adsorption capacity for water treatment design Given Langmuir equilibrium constant (b) = 1.9880 L/mg; equilibrium concentration (C_e) = 42.7000 mg/L; adsorption capacity (x/m) (x_m) = 65.0000 mg/g, determine the maximum adsorption capacity (a) in mg/g.

Start with the thinking

• The governing relation printed in this handbook section is Langmuir Isotherm.
• Everything except a is given, so isolate a 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 Langmuir isotherm describes monolayer adsorption capacity and can be expressed in a linearized form for parameter fitting.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\dfrac{x}{m} = \dfrac{a b C_e}{1 + b C_e}$$

2. Step 2 — Rearrange symbolically for a:

   $$a = \dfrac{(x/m)(1+bC_e)}{bC_e}$$

3. Step 3 — List the givens: Langmuir equilibrium constant (b) = 1.9880 L/mg, equilibrium concentration (C_e) = 42.7000 mg/L, adsorption capacity (x/m) (x_m) = 65.0000 mg/g.

4. Step 4 — Substitute the given values:

   $$a = \dfrac{(x/m)(1+bC_e)}{bC_e}$$

5. Step 5 — Evaluate:

   $$a = 65.7657\ \text{mg/g}$$

6. Step 6 — Check: returning a = 65.7657 mg/g to

   $$\dfrac{x}{m} = \dfrac{a b C_e}{1 + b C_e}$$

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

Why the other options are there

• 131.5 — kept a factor of two that cancels in the correct rearrangement.
• 32.8829 — dropped that same factor in the other direction.
• 72.3423 — rounded an intermediate value before the final step.

Reference: FE Handbook — Langmuir Isotherm

Langmuir isotherm monolayer adsorption model for carbon treatment Given maximum adsorption capacity (a) = 108.0 mg/g; Langmuir equilibrium constant (b) = 0.8500 L/mg; equilibrium concentration (C_e) = 2.1000 mg/L, determine the adsorption capacity (x/m) (x_m) in mg/g.

Start with the thinking

• The governing relation printed in this handbook section is Langmuir 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 Langmuir isotherm describes monolayer adsorption capacity and can be expressed in a linearized form for parameter fitting.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\dfrac{x}{m} = \dfrac{a b C_e}{1 + b C_e}$$

2. Step 2 — Rearrange symbolically for x_m:

   $$x_{m} = \dfrac{abC_e}{1+bC_e}$$

3. Step 3 — List the givens: maximum adsorption capacity (a) = 108.0 mg/g, Langmuir equilibrium constant (b) = 0.8500 L/mg, equilibrium concentration (C_e) = 2.1000 mg/L.

4. Step 4 — Substitute the given values:

   $$x_{m} = \dfrac{abC_e}{1+bC_e}$$

5. Step 5 — Evaluate:

   $$x_{m} = 69.2208\ \text{mg/g}$$

6. Step 6 — Check: returning x_m = 69.2208 mg/g to

   $$\dfrac{x}{m} = \dfrac{a b C_e}{1 + b C_e}$$

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

Why the other options are there

• 138.4 — kept a factor of two that cancels in the correct rearrangement.
• 34.6104 — dropped that same factor in the other direction.
• 76.1429 — rounded an intermediate value before the final step.

Reference: FE Handbook — Langmuir Isotherm

Langmuir isotherm linearized form used to determine adsorption constants Given maximum adsorption capacity (a) = 133.0 mg/g; Langmuir equilibrium constant (b) = 0.1920 L/mg; adsorption capacity (x/m) (x_m) = 102.7 mg/g, determine the equilibrium concentration (C_e) in mg/L.

Start with the thinking

• The governing relation printed in this handbook section is Langmuir 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 Langmuir isotherm describes monolayer adsorption capacity and can be expressed in a linearized form for parameter fitting.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\dfrac{x}{m} = \dfrac{a b C_e}{1 + b C_e}$$

2. Step 2 — Rearrange symbolically for C_e:

   $$C_{e} = \dfrac{x/m}{ab-(x/m)b}$$

3. Step 3 — List the givens: maximum adsorption capacity (a) = 133.0 mg/g, Langmuir equilibrium constant (b) = 0.1920 L/mg, adsorption capacity (x/m) (x_m) = 102.7 mg/g.

4. Step 4 — Substitute the given values:

   $$C_{e} = \dfrac{x/m}{a0.1920-(x/m)0.1920}$$

5. Step 5 — Evaluate:

   $$C_{e} = 17.6533\ \text{mg/L}$$

6. Step 6 — Check: returning C_e = 17.6533 mg/L to

   $$\dfrac{x}{m} = \dfrac{a b C_e}{1 + b C_e}$$

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

Why the other options are there

• 35.3067 — kept a factor of two that cancels in the correct rearrangement.
• 8.8267 — dropped that same factor in the other direction.
• 19.4187 — rounded an intermediate value before the final step.

Reference: FE Handbook — Langmuir Isotherm

Langmuir isotherm adsorption capacity for water treatment design Given Langmuir equilibrium constant (b) = 0.7660 L/mg; equilibrium concentration (C_e) = 11.4000 mg/L; adsorption capacity (x/m) (x_m) = 27.8000 mg/g, determine the maximum adsorption capacity (a) in mg/g.

Start with the thinking

• The governing relation printed in this handbook section is Langmuir Isotherm.
• Everything except a is given, so isolate a 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 Langmuir isotherm describes monolayer adsorption capacity and can be expressed in a linearized form for parameter fitting.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\dfrac{x}{m} = \dfrac{a b C_e}{1 + b C_e}$$

2. Step 2 — Rearrange symbolically for a:

   $$a = \dfrac{(x/m)(1+bC_e)}{bC_e}$$

3. Step 3 — List the givens: Langmuir equilibrium constant (b) = 0.7660 L/mg, equilibrium concentration (C_e) = 11.4000 mg/L, adsorption capacity (x/m) (x_m) = 27.8000 mg/g.

4. Step 4 — Substitute the given values:

   $$a = \dfrac{(x/m)(1+bC_e)}{bC_e}$$

5. Step 5 — Evaluate:

   $$a = 30.9835\ \text{mg/g}$$

6. Step 6 — Check: returning a = 30.9835 mg/g to

   $$\dfrac{x}{m} = \dfrac{a b C_e}{1 + b C_e}$$

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

Why the other options are there

• 61.9671 — kept a factor of two that cancels in the correct rearrangement.
• 15.4918 — dropped that same factor in the other direction.
• 34.0819 — rounded an intermediate value before the final step.

Reference: FE Handbook — Langmuir Isotherm

Langmuir isotherm monolayer adsorption model for carbon treatment Given maximum adsorption capacity (a) = 100.0 mg/g; Langmuir equilibrium constant (b) = 1.7410 L/mg; equilibrium concentration (C_e) = 0.9000 mg/L, determine the adsorption capacity (x/m) (x_m) in mg/g.

Start with the thinking

• The governing relation printed in this handbook section is Langmuir 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 Langmuir isotherm describes monolayer adsorption capacity and can be expressed in a linearized form for parameter fitting.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\dfrac{x}{m} = \dfrac{a b C_e}{1 + b C_e}$$

2. Step 2 — Rearrange symbolically for x_m:

   $$x_{m} = \dfrac{abC_e}{1+bC_e}$$

3. Step 3 — List the givens: maximum adsorption capacity (a) = 100.0 mg/g, Langmuir equilibrium constant (b) = 1.7410 L/mg, equilibrium concentration (C_e) = 0.9000 mg/L.

4. Step 4 — Substitute the given values:

   $$x_{m} = \dfrac{abC_e}{1+bC_e}$$

5. Step 5 — Evaluate:

   $$x_{m} = 61.0425\ \text{mg/g}$$

6. Step 6 — Check: returning x_m = 61.0425 mg/g to

   $$\dfrac{x}{m} = \dfrac{a b C_e}{1 + b C_e}$$

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

Why the other options are there

• 122.1 — kept a factor of two that cancels in the correct rearrangement.
• 30.5213 — dropped that same factor in the other direction.
• 67.1468 — rounded an intermediate value before the final step.

Reference: FE Handbook — Langmuir Isotherm