Question

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

Answer 1

Hint: We can solve this problem by using the Debye-Huckel Limiting Law, which relates the mean activity coefficient with the ionic strength (I). The ionic strength is the summation of the product of the concentration (molal) and the charge on the respective disintegrated ion.

Complete Step By Step Answer:
The Debye-Huckel Theory gives us an equation for calculating the mean activity coefficient denoted by $ \gamma \pm $ for solutions having concentrations less than or equal to 0.01 M. The equation can be given as:
The Debye-Huckel Theory gives us an equation for calculating the mean activity coefficient denoted by $ \gamma \pm $ for solutions having concentrations less than or equal to 0.01 M. The equation can be given as:
 $ \log \gamma \pm = - \dfrac{{A\sqrt I }}{{1 + Ba\sqrt I }} \times |{Z^ + }{Z^ - }| $
Where, $ {Z^ + },{Z^ - } $ is the charge on the cation and anion respectively T is the temperature and I is the ionic strength. The value of A can be given by the formula: $ A = {(2\pi {N_A}{\rho _A})^{1/2}}{\left( {\dfrac{{{e^2}}}{{4\pi {\varepsilon _0}{\varepsilon _{r,A}}{k_B}T}}} \right)^{3/2}} $
Where $ {\varepsilon _0},{\varepsilon _r} $ is the permeability in space and relative permeability. The constant B is given by the formula: $ B = e{\left( {\dfrac{{2{N_A}{\rho _A}}}{{{\varepsilon _0}{\varepsilon _{r.A}}{k_B}T}}} \right)^{1/2}} $
The dielectric constant for methanol is 32.6 and the temperature given to us is $ {25^ \circ }C $ = 298 K.
Substituting the respective values to find the values of A and B we get them as, $ A = 3.8885{(kg/mol)^{1/2}},B = 4.5245 \times {10^9}{(kg/mol)^{1/2}}{m^{ - 1}} $
The derivation of A and B isn’t important.
To find the mean activity coefficient we’ll first have to find out the ionic strength of the 0.0200 molal HCl solution. The ionic strength can be given by the formula: $ I = \dfrac{1}{2}\sum {{m_i}{z_i}^2} $
Where m is the concentration of the respective ion in molal and z is the charge on that ion.
The disintegration of the compound given to us can hereby happen like this:
 $ HCl \to {H^ + } + C{l^ - } $
The charge on H is +1 and that on Cl is -1. The ionic strength therefore can be calculated as:
 $ I = \dfrac{1}{2}[(0.0200){( + 1)^2} + (0.0200){( - 1)^2}] $
 $ I = \dfrac{1}{2}[0.020 + 0.020] $
 $ I = \dfrac{1}{2}[0.040] = 0.020m $
Now that we know the ionic strength I we can calculate the mean activity coefficient $ \gamma \pm $ by substituting the respective values.
The information given to us is:
 $ \varepsilon = 32.6,T = 298K,{Z^ + } = + 1,{Z^ - } = - 1,I = 0.020m $
Substituting the values in the equation above:
 $ \log \gamma \pm = - \dfrac{{3.8885 \times \sqrt {0.020} }}{{1 + (4.5245 \times {{10}^9})(3 \times {{10}^{ - 10}})(\sqrt {0.020)} }} \times | + 1 \times - 1| $
 $ \log \gamma \pm = - 0.4614 $
To find the value of $ \gamma \pm $ we’ll take the antilog on both sides.
 $ \gamma \pm = AL( - 0.4614) = 0.6304 $
This is the required answer.

Note:
Since the charges do not have any units, the units of activity and mean activity coefficients are the same as that of concentrations i.e. molal. If we are told to find only the mean activity coefficient we can stop right after finding the values for $ \gamma \pm $ .