Write the Lippmann capillary equation and how do you determine charge density from it ?
The Lippmann Capillary Equation
The Lippmann equation describes the relationship between the surface tension of an electrode interface and the electrical potential applied to it. It is fundamental in electrochemistry for understanding the structure of the electrical double layer.
1. The Equation
The equation is derived from the Gibbs adsorption isotherm for a mercury drop electrode (or similar ideal polarized electrodes). It is expressed as:
$$\left( \frac{\partial \gamma}{\partial E} \right)_{T, P, \mu} = -\sigma_M$$
Where:
- $\gamma$: The surface tension (or surface excess free energy).
- $E$: The applied potential.
- $\sigma_M$: The surface charge density on the metal electrode.
- The subscripts $T, P, \mu$ indicate that temperature, pressure, and chemical potential (composition) are held constant.
2. Determining Charge Density ($\sigma_M$)
The surface charge density is determined experimentally by observing how the surface tension of a liquid metal electrode (like the Dropping Mercury Electrode) changes with potential. This process involves the following steps:
A. Electrocapillary Curve Measurement
Experimentally, you measure the surface tension ($\gamma$) as a function of the applied potential ($E$). When plotted, this results in a parabolic curve known as the electrocapillary curve.
B. Differentiation
According to the Lippmann equation, the negative slope of the electrocapillary curve at any given potential provides the surface charge density at that point:
The slope of the tangent to the $\gamma$ vs. $E$ curve at potential $E_1$ is equal to the negative of the charge density $\sigma_M$ at that potential.
C. Identifying the PZC
At the peak of the parabola (where the slope is zero), the charge density is zero. This point is known as the Potential of Zero Charge (PZC). $$\frac{\partial \gamma}{\partial E} = 0 \implies \sigma_M = 0$$
- Left of PZC: The slope is positive, meaning $\sigma_M$ is negative (excess electrons).
- Right of PZC: The slope is negative, meaning $\sigma_M$ is positive (electron deficiency).
3. Extension: Differential Capacitance
If you differentiate the Lippmann equation a second time with respect to potential, you can determine the differential capacitance ($C_d$) of the double layer:
$$\left( \frac{\partial^2 \gamma}{\partial E^2} \right) = -\frac{\partial \sigma_M}{\partial E} = -C_d$$
This shows that the curvature of the electrocapillary plot is directly related to the ability of the interface to store charge.
Summary of Relationships
| Observation | Mathematical Deduction | Physical State of Electrode |
|---|---|---|
| Zero Slope (Peak) | σM = 0 | Uncharged (PZC) |
| Positive Slope | σM < 0 | Negatively Charged |
| Negative Slope | σM > 0 | Positively Charged |