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Diffusion of A through stagnant B in a liquid. The solute hcl (A) is diffusing through a thin film of water (B) 2.0 mm thick at 283 K. The concentration of HCl at point 1 at one boundary of the film is 12.0 wt% HCl (density1 is 1060.7 kg/m3), and at the other boundary at point 2 it is 6.0 wt% HCl (density2 is 1030.3 kg/m3). The diffusion coefficient of HCl in water is 2.5x10-9 m2/s. Assuming steady state and one boundary impermeable to water, calculate the flux of HCl in kg mol/s.m2. How to solve it?

Sagot :

Assuming steady state and one boundary impermeable to water, the flux of HCl is; Nₐ = 2.355 × 10⁻⁶ kmol/m.s²

We are given;

Density of Solution at Point 1; ρ₁ = 1060.7 Kg/m³

Density of Solution at Point 2; ρ₂ = 1030.3 kg/m³

Diffusion coefficient of HCl in water; D_ab = 2.5 × 10⁻⁹ m²/s

Thickness of thin film; Z = 2 mm = 0.002 m

From tables;

Molar mass of HCl; M_a = 36.5 kg/mol

Molar mass of H₂O; M_b = 18 kg/mol

At Point 1;

Let us calculate mole fraction of HCl;

[tex]x_{A1} = \frac{\frac{12}{36.5}}{\frac{12}{36.5} + \frac{88}{18}}[/tex]

⇒ [tex]x_{A1} = 0.0629[/tex]

[tex]x_{B1} = 1 - 0.0629\\\\x_{B1} = 0.9371[/tex]

Average molecular weight here is;

[tex]M_{1} = \frac{{100}}{\frac{12}{36.5} + \frac{88}{18}}[/tex]

⇒ [tex]M_{1} = 19.1662 kg/kmol[/tex]

At Point 2;

Let us calculate mole fraction of HCl;

[tex]x_{A2} = \frac{\frac{6}{36.5}}{\frac{6}{36.5} + \frac{94}{18}}[/tex]

⇒ [tex]x_{A2} = 0.0305[/tex]

[tex]x_{B2} = 1 - 0.0305\\\\x_{B2} = 0.9695[/tex]

Average molecular weight here is;

[tex]M_{2} = \frac{{100}}{\frac{6}{36.5} + \frac{94}{18}}[/tex]

⇒ [tex]M_{2} = 18.5646 kg/kmol[/tex]

Average concentration is;

C_av = ¹/₂[(ρ₁/M₁) + ρ₂/M₂)]

C_av = ¹/₂((1060.7/19.1662) + (1030.3/18.5646))

C_av = 55.42 kmol/m³

x_bm = (x_b2 - x_b1)/(In(x_b2/x_b1))

x_bm = (0.9695 - 0.9371)/(In (0.9695/0.9371))

x_bm = 0.9529

Thus, flux of HCl  is;

Nₐ = [D_ab × C_av × (x_a1 - x_a2)]/(x_bm * Z)

Plugging in the relevant values gives;

Nₐ = [2.5 × 10⁻⁹ × 55.42 × (0.0629 - 0.0305)]/(0.9529 × 0.002)

Nₐ = 2.355 × 10⁻⁶ kmol/m.s²

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