Use the Debye-Huckel equation to evaluate $γ_{\pm}$ $gamma_(pm)$ for a $0.0200 mol/kg$ $"0.0200 mol/kg"$ $HCl$ $"HCl"$ solution, with methanol as the solvent, at $25^{\circ} C$ $25^@ "C"$ and $1 atm$ $"1 atm"$ ?

Question

Use the Debye-Huckel equation to evaluate $γ_{\pm}$ $gamma_(pm)$ for a $0.0200 mol/kg$ $"0.0200 mol/kg"$ $HCl$ $"HCl"$ solution, with methanol as the solvent, at $25^{\circ} C$ $25^@ "C"$ and $1 atm$ $"1 atm"$ ?

The answer is $γ_{\pm} = 0.6304$ $gamma_(pm) = 0.6304$ . I am writing this answer because this equation is rather difficult to evaluate correctly.

The density of methanol at this $T$ $T$ and $P$ $P$ is ${0.787 g/cm}^{3}$ $"0.787 g/cm"^3$ , while its dielectric constant is $ε_{r} = 32.6$ $epsilon_r = 32.6$ .

Main Equation:

${ln γ}_{\pm} = - | z_{+} z_{-} | \frac{A I_{m}^{1/2}}{1 + B a I_{m}^{1/2}}$ $ln gamma_(pm) = -|z_(+)z_(-)|(A I_m^"1/2")/(1 + BaI_m^"1/2")$

Sub-Equations:

$A = {(2 π N_{A} ρ_{A})}_{A}^{1/2} {(\frac{e^{2}}{4 π ε_{0} ε_{r, A} k_{B} T})}^{3/2}$ $A = (2piN_A rho_A)^"1/2"(e^2/(4piepsilon_0 epsilon_(r,A)k_BT))^"3/2"$

$B = e {(\frac{2 N_{a} ρ_{A}}{ε_{0} ε_{r, A} k_{B} T})}^{1/2}$ $B = e((2N_arho_A)/(epsilon_0 epsilon_(r,A) k_B T))^"1/2"$

Molal-scale Ionic Strength:
$I_m = sum_i z_i^2 m_i = stackrel("If single strong electrolyte")overbrace(1/2 |z_(+)z_(-)|(nu_(+) + nu_(-)) m_i)$

Assume $a ~~ 3xx10^(-10) "m"$ , or 3 angstroms, the mean ionic radius (including the hydration sphere).

Definitions:

$gamma_(pm)$ is the activity coefficient of a solution containing some set of strong electrolytes, ignoring ion pairing interactions.

$nu_(+)$ ( $nu_(-)$ ) is the stoichiometric coefficient of the cation (anion).

$z_(-)$ ( $z_(+)$ ) is the charge of the anion (cation).

$m_i$ is the molality of the solute in $"mol/kg"$ .

$N_A$ is Avogadro's number, $6.022xx10^(23) "mol"^(-1)$ .

$rho_A$ is the density of the solvent in $"kg/m"^3$ .

$epsilon_0 = 8.854xx10^(-12)$ $"C"^2"/N"cdot"m"^2$ is the vacuum permittivity.

$epsilon_(r,A)$ is the dielectric constant of the solvent, with no units.

$k_B = 1.3807xx10^(-23)$ $"J/K"$ is the Boltzmann constant.

$T$ is the temperature in $"K"$ .

$e$ is the proton charge, $1.602xx10^(-19)$ $"C"$

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Answer 1 · 2016-11-21T09:31:24

From the definitions above, I will evaluate $A$ and $B$ , followed by $I_m$ , and then $ln gamma_(pm)$ , and then $gamma_(pm)$ .

Sub-Equation Evaluations

$color(green)(A) = (2piN_A rho_A)^"1/2"(e^2/(4piepsilon_0 epsilon_(r,A)k_BT))^"3/2"$

$= (2pi(6.022xx10^(23) "mol"^(-1))("787 kg/m"^3))^"1/2"$
$[(1.602xx10^(-19) "C")^2/(4pi(8.854xx10^(-12) "C"^2"/N"cdot"m"^2)(32.6)(1.3807xx10^(-23) "J/K")("298.15 K"))]^"3/2"$

$= color(green)(3.8885)$ $color(green)(("kg"/"mol")^"1/2")$

$color(green)(B) = e((2N_Arho_A)/(epsilon_0 epsilon_(r,A) k_BT))^"1/2"$

$= (1.602xx10^(-19) "C")$
$[(2(6.022xx10^(23) "mol"^(-1))("787 kg/m"^3))/((8.854xx10^(-12) "C"^2"/N"cdot"m"^2)(32.6)(1.3807xx10^(-23) "J/K")("298.15 K"))]^"1/2"$

$= color(green)(4.5245xx10^9)$ $color(green)(("kg"/"mol")^"1/2" "m"^(-1))$

Now that those crazy ones are out of the way, here's the fairly easy one. The molal-scale ionic strength is the measure of total ion concentration, basically. Here's how you could use the equation:

$color(green)(I_m) = 1/2sum_i z_i^2 m_i$

$= 1/2[z_(+)^2m_(+) + z_(-)^2 m_(-)]$

$= 1/2[z_(+)^2 nu_(+)m_i + z_(-)^2nu_(-)m_i]$

$= 1/2[|z_(+)z_(-)|nu_(+)m_i + |z_(+)z_(-)|nu_(-)m_i]$

$= 1/2|z_(+)z_(-)|(nu_(+) + nu_(-))m_i$

$= 1/2|1*-1|(1+1)("0.0200 mol HCl"/"kg methanol") = color(green)("0.0200 mol HCl"/"kg methanol")$

Main Equation Evaluation

Now that those are calculated, we can use them in the main equation to get $gamma_(pm)$ . Again, assuming $a ~~ 3xx10^(-10) "m"$ .

$ln gamma_(pm) = -|z_(+)z_(-)|(A I_m^"1/2")/(1 + BaI_m^"1/2")$

$= -|1*1|$
$[(3.8885 ("kg"/"mol")^"1/2"("0.0200 mol"/"kg")^"1/2")/(1 + (4.5245xx10^9 ("kg"/"mol")^"1/2" "m"^(-1))(3xx10^(-10) "m")("0.0200 mol"/"kg")^"1/2")]$

$= -0.4614$

Therefore, we finally have that:

$color(blue)(gamma_(pm)) = e^(ln gamma_(pm)) = color(blue)(0.6304)$

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Use the Debye-Huckel equation to evaluate $γ_{\pm}$ $gamma_(pm)$ for a $0.0200 mol/kg$ $"0.0200 mol/kg"$ $HCl$ $"HCl"$ solution, with methanol as the solvent, at $25^{\circ} C$ $25^@ "C"$ and $1 atm$ $"1 atm"$ ?

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Use the Debye-Huckel equation to evaluate $γ_{\pm}$ $gamma_(pm)$ for a $0.0200 mol/kg$ $"0.0200 mol/kg"$ $HCl$ $"HCl"$ solution, with methanol as the solvent, at $25^{\circ} C$ $25^@ "C"$ and $1 atm$ $"1 atm"$ ?