The electrolyte solution that has the smallest Debye-length at 298 K is
Understanding Debye-length
The Debye-length ($\lambda_D$) is a fundamental parameter in the theory of electrolyte solutions. It represents the characteristic distance over which electrostatic interactions between ions are screened by the surrounding ionic atmosphere. In simpler terms, it indicates the thickness of the diffuse layer of counterions that surrounds a central ion. A smaller Debye-length means that the ionic atmosphere is more compact, and the shielding of the central ion's charge is more effective over shorter distances.
The Debye-length is inversely proportional to the square root of the ionic strength ($I$) of the solution. The relationship is given by the formula:
$$ \lambda_D \propto \frac{1}{\sqrt{I}} $$
Therefore, the electrolyte solution with the highest ionic strength will have the smallest Debye-length at a given temperature.
Calculating Ionic Strength
Ionic strength ($I$) is a measure of the total concentration of ions in a solution, accounting for the charge of each ion. It is defined by the equation:
$$ I = \frac{1}{2} \sum_i c_i z_i^2 $$
where $c_i$ is the molar concentration of ion $i$ and $z_i$ is its charge number.
We need to calculate the ionic strength for each of the given 0.01 M electrolyte solutions:
| Electrolyte | Dissociation | Ion Concentrations | Ion Charges | Ionic Strength ($I$) Calculation | Calculated $I$ |
|---|---|---|---|---|---|
| 0.01 M NaCl | Na$^+$ + Cl$^-$ | cNa⁺ = 0.01 M cCl⁻ = 0.01 M |
zNa⁺ = +1 zCl⁻ = -1 |
$$I = \frac{1}{2} [(0.01)(+1)^2 + (0.01)(-1)^2]$$ $$= \frac{1}{2} [0.01 + 0.01] = \frac{1}{2} [0.02]$$ | 0.01 M |
| 0.01 M Na₂SO₄ | 2Na$^+$ + SO₄²⁻ | cNa⁺ = 2 × 0.01 = 0.02 M cSO₄²⁻ = 0.01 M |
zNa⁺ = +1 zSO₄²⁻ = -2 |
$$I = \frac{1}{2} [(0.02)(+1)^2 + (0.01)(-2)^2]$$ $$= \frac{1}{2} [0.02 + 0.01 \times 4] = \frac{1}{2} [0.02 + 0.04]$$ | 0.03 M |
| 0.01 M CuCl₂ | Cu²⁺ + 2Cl$^-$ | cCu²⁺ = 0.01 M cCl⁻ = 2 × 0.01 = 0.02 M |
zCu²⁺ = +2 zCl⁻ = -1 |
$$I = \frac{1}{2} [(0.01)(+2)^2 + (0.02)(-1)^2]$$ $$= \frac{1}{2} [0.01 \times 4 + 0.02] = \frac{1}{2} [0.04 + 0.02]$$ | 0.03 M |
| 0.01 M LaCl₃ | La³⁺ + 3Cl$^-$ | cLa³⁺ = 0.01 M cCl⁻ = 3 × 0.01 = 0.03 M |
zLa³⁺ = +3 zCl⁻ = -1 |
$$I = \frac{1}{2} [(0.01)(+3)^2 + (0.03)(-1)^2]$$ $$= \frac{1}{2} [0.01 \times 9 + 0.03] = \frac{1}{2} [0.09 + 0.03]$$ | 0.06 M |
Comparing Ionic Strengths
Comparing the calculated ionic strengths:
- 0.01 M NaCl: I = 0.01 M
- 0.01 M Na₂SO₄: I = 0.03 M
- 0.01 M CuCl₂: I = 0.03 M
- 0.01 M LaCl₃: I = 0.06 M
The ionic strengths are in the order: I(NaCl) < I(Na₂SO₄) = I(CuCl₂) < I(LaCl₃).
Determining Smallest Debye-length
Since the Debye-length is inversely proportional to the square root of the ionic strength, the solution with the highest ionic strength will have the smallest Debye-length.
From our calculations, 0.01 M LaCl₃ has the highest ionic strength (0.06 M).
Therefore, 0.01 M LaCl₃ has the smallest Debye-length among the given options at 298 K.
Important Questions from Debye Huckel Theory
The amount of Ba(NO3)2 (molecular weight 261.32 amu) required to be added to 500 g of a 0.11 mol kg-1 solution of KNO3 in order to raise its ionic strength to 1.00 is approximately:
- Debye - Huckel limiting law in :
- The mean ionic activity coefficient of 0.001 molal $ZnSO_4$ (aq) at 298 K according to the Debye- Hückel limiting law is (Debye-Hückel constant is 0.509 $molal^{-½}$) ________
- The ionic activity coefficients of $Ca^{2+}$ and $F^-$ are 0.72 and 0.28, respectively. The mean activity coefficient of $CaF_2$ is ________ (up to two decimal places)
- For a 1 molal aqueous NaCl solution, the mean ionic activity coefficient ($\gamma_\pm$) and the Debye- Hückel Limiting Law constant (A) are related as
