Does debye huckel equation explain about non ideal behavior of solution ?
Answers
The chemists Peter Debye and Erich Hückel noticed that solutions that contain ionic solutes do not behave ideally even at very low concentrations. So, while the concentration of the solutes is fundamental to the calculation of the dynamics of a solution, they theorized that an extra factor that they termed gamma is necessary to the calculation of the activity coefficients of the solution. Hence they developed the Debye–Hückel equation and Debye–Hückel limiting law. The activity is only proportional to the concentration and is altered by a factor known as the activity coefficient {\displaystyle \ \gamma \,} \ \gamma \,. This factor takes into account the interaction energy of ions in the solution.In order to calculate the activity, {\displaystyle \ a_{C}\,} \ a_{C}\,, of an ion, C, in a solution, one must know the concentration and the activity coefficient:
{\displaystyle \ a_{C}=\gamma {\frac {[C]}{[C^{\ominus }]}}\,} \ a_{C}=\gamma {\frac {[C]}{[C^{\ominus }]}}\,
where
{\displaystyle \ \gamma \,} \ \gamma \, is the activity coefficient of C
{\displaystyle \ [C^{\ominus }]\,} \ [C^{\ominus }]\, is the concentration of the chosen standard state, e.g. 1 mol/kg if molality is used.
{\displaystyle [C]} [C] is a measure of the concentration of C
Dividing {\displaystyle [C]} [C] with {\displaystyle [C^{\ominus }]} [C^{\ominus }] gives a dimensionless quantity
The Debye–Hückel limiting law enables one to determine the activity coefficient of an ion in a dilute solution of known ionic strength. The equation is[1]:section 2.5.2
{\displaystyle \ln(\gamma _{i})=-{\frac {z_{i}^{2}q^{2}\kappa }{8\pi \varepsilon _{r}\varepsilon _{0}k_{B}T}}=-{\frac {z_{i}^{2}q^{3}N_{\mathrm {A} }^{1/2}}{4\pi (\varepsilon _{r}\varepsilon _{0}k_{B}T)^{3/2}}}{\sqrt {\frac {I}{2}}}=-Az_{i}^{2}{\sqrt {I}}} \ln(\gamma _{i})=-{\frac {z_{i}^{2}q^{2}\kappa }{8\pi \varepsilon _{r}\varepsilon _{0}k_{B}T}}=-{\frac {z_{i}^{2}q^{3}N_{\mathrm {A} }^{1/2}}{4\pi (\varepsilon _{r}\varepsilon _{0}k_{B}T)^{3/2}}}{\sqrt {\frac {I}{2}}}=-Az_{i}^{2}{\sqrt {I}}
{\displaystyle z_{i}} z_{i} is the charge number of ion species i
{\displaystyle q} q is the elementary charge
{\displaystyle \kappa } \kappa is the inverse of the Debye screening length, defined below
{\displaystyle \varepsilon _{r}} \varepsilon _{r} is the relative permittivity of the solvent
{\displaystyle \varepsilon _{0}} \varepsilon _{0} is the permittivity of free space
{\displaystyle k_{B}} k_{B} is Boltzmann's constant
{\displaystyle T} T is the temperature of the solution
{\displaystyle N_{\mathrm {A} }} N_{\mathrm {A} } is Avogadro's number
{\displaystyle I} I is the ionic strength of the solution, defined below
{\displaystyle A} A is a constant that depends on temperature. If {\displaystyle I} I is expressed in terms of molality, instead of molarity (as in the equation above and in the rest of this article), then an experimental value for {\displaystyle A} A of water is {\displaystyle 1.172{\text{ mol}}^{-1/2}{\text{kg}}^{1/2}} {\displaystyle 1.172{\text{ mol}}^{-1/2}{\text{kg}}^{1/2}} at 25 C. It is common to use a base-10 logarithm, in which case we factor {\displaystyle \ln 10} \ln 10, so A is {\displaystyle 0.509{\text{ mol}}^{-1/2}{\text{kg}}^{1/2}} {\displaystyle 0.509{\text{ mol}}^{-1/2}{\text{kg}}^{1/2}}.
It is important to note that because the ions in the solution act together, the activity coefficient obtained from this equation is actually a mean activity coefficient.
The excess osmotic pressure which is obtained from Debye–Hückel theory is in cgs units:[1]
{\displaystyle P^{\mathrm {ex} }=-{\frac {k_{B}T\kappa _{\mathrm {cgs} }^{3}}{24\pi }}=-{\frac {k_{B}T\left({\frac {4\pi \sum _{j}c_{j}q_{j}}{\epsilon _{0}\epsilon _{r}k_{B}T}}\right)^{3/2}}{24\pi }}} {\displaystyle P^{\mathrm {ex} }=-{\frac {k_{B}T\kappa _{\mathrm {cgs} }^{3}}{24\pi }}=-{\frac {k_{B}T\left({\frac {4\pi \sum _{j}c_{j}q_{j}}{\epsilon _{0}\epsilon _{r}k_{B}T}}\right)^{3/2}}{24\pi }}}.
Therefore, the total pressure is the sum of the excess osmotic pressure and the ideal pressure {\displaystyle P^{\mathrm {id} }=k_{B}T\sum _{i}c_{i}} {\displaystyle P^{\mathrm {id} }=k_{B}T\sum _{i}c_{i}}. The osmotic coefficient is then given by:
{\displaystyle \phi ={\frac {P^{\mathrm {id} }+P^{\mathrm {ex} }}{P^{\mathrm {id} }}}=1+{\frac {P^{\mathrm {ex} }}{P^{\mathrm {id} }}}} {\displaystyle \phi ={\frac {P^{\mathrm {id} }+P^{\mathrm {ex} }}{P^{\mathrm {id} }}}=1+{\frac {P^{\mathrm {ex} }}{P^{\mathrm {id} }}}}