Question

One of the factors contributing to the high relative permittivity of water to static electric fields is the permanent electric dipole moment \(p\) of the water molecule. In an external field \(E\) the dipoles tend to line up with the field, but they do not do so completely because of thermal agitation at the temperature \(T\) of the water. A classical (non-quantum) calculation using the Boltzmann distribution shows that the average polarisability per molecule \(\alpha\) is given by $$ \alpha=\frac{p}{E}\left(\operatorname{coth} x-x^{-1}\right) $$ where \(x=p E / k T\) and \(k\) is the Boltzmann constant. At ordinary temperatures, even with high field strengths \(\left(10^{4} \mathrm{Vm}^{-1}\right.\) or more), \(x \ll 1\). By making suitable series expansions of the hyperbolic functions involved, show that \(\alpha=p^{2} / 3 k T\) to an accuracy of about one part in \(15 x^{-2}\).

Short answer

The average polarizability per molecule \(\alpha = \frac{p^2}{3 k T}\), with accuracy to about one part in \(15x^{-2}\).

Step-by-step solution

Understand the Given Formula

The problem provides the formula for average polarizability per molecule: \(\alpha=\frac{p}{E}\left(\operatorname{coth} x-x^{-1}\right)\)Where \(x=\frac{p E}{k T}\), and it is given that \(x \ll 1\).

Series Expansion for Hyperbolic Cotangent

Recall that the hyperbolic cotangent function is defined as \(\operatorname{coth} x = \frac{\cosh x}{\sinh x}\).For small \(x\), use the Taylor series expansions:\(\sinh x \approx x + \frac{x^3}{6}\) and\(\cosh x \approx 1 + \frac{x^2}{2}\).

Simplify the Hyperbolic Cotangent

Using the series expansions in Step 2,\(\operatorname{coth} x = \frac{1 + \frac{x^2}{2}}{x + \frac{x^3}{6}} \approx \frac{1}{x} + \frac{x}{3}\).Thus, for small \(x\): \(\operatorname{coth} x \approx \frac{1}{x} + \frac{x}{3}\).

Substitute Back into the Polarizability Formula

Substitute \(\operatorname{coth} x \approx \frac{1}{x} + \frac{x}{3}\) into the given polarizability formula:\(\alpha = \frac{p}{E}\left(\frac{1}{x} + \frac{x}{3} - \frac{1}{x}\right) \).The \(\frac{1}{x}\) terms cancel each other, leaving\(\alpha = \frac{p}{E} \cdot \frac{x}{3}\).

Substitute and Simplify Further

Recall that \(x = \frac{p E}{k T}\). Substitute this into the simplified formula:\(\alpha = \frac{p}{E} \cdot \frac{\frac{p E}{k T}}{3} \).This simplifies to:\(\alpha = \frac{p^2}{3 k T}\).

Accuracy Analysis

The series expansion of \(\operatorname{coth} x\) included terms up to \(x\). For higher accuracy, the next term in the series expansion would be of order \(x^{-2}\). Hence, the correction term would be on the order of about \(\frac{1}{15x^2}\).

Key concepts

Boltzmann Distribution

The Boltzmann distribution is crucial in understanding how particles distribute themselves among various energy states in a system at thermal equilibrium. It indicates that the probability of a particle having energy \(E\) is proportional to \(\text{exp}(-E / kT)\), where \(k\) is the Boltzmann constant and \(T\) is the temperature.
This concept plays an essential role in our exercise since it helps describe how electric dipoles, like water molecules, react to external electric fields while influenced by thermal motion. When we say that dipoles align according to the Boltzmann distribution, we mean higher-energy alignments are less probable than lower-energy ones, affected by \(E\) and \(T\). The temperature \(T\) influences thermal agitation, affecting how well the dipoles can align with the field.

Hyperbolic Functions

Hyperbolic functions like \(\text{coth} x\) (hyperbolic cotangent) are analogous to trigonometric functions but for a hyperbola rather than a circle. In our exercise, hyperbolic cotangent is defined as \(\text{coth} x = \frac{\text{cosh} x}{\text{sinh} x}\), where \(\text{cosh} x\) and \(\text{sinh} x\) are the hyperbolic cosine and sine functions, respectively.

Using these functions helps describe non-linear effects in electric fields. For small values of \(x\), we use the Taylor series expansions to approximate these functions. For instance, \(\text{sinh} x \thickapprox x + \frac{x^3}{6}\) and \(\text{cosh} x \thickapprox 1 + \frac{x^2}{2}\) explain behavior for \(x \to 0\). These approximations simplify our formula for polarizability.

Series Expansion

Series expansion is a mathematical method that allows us to approximate complex functions with simpler polynomial terms. This technique is particularly useful when dealing with small values. In this exercise, we expand the hyperbolic functions \(\text{sinh} x\) and \(\text{cosh} x\):

• \(\text{sinh} x \thickapprox x + \frac{x^3}{6}\)
• \(\text{cosh} x \thickapprox 1 + \frac{x^2}{2}\)

These expansions make it feasible to derive \(\text{coth} x\) for small \(x\) as \(\text{coth} x \thickapprox \frac{1}{x} + \frac{x}{3}\).
Using series expansion allows us to simplify the problem and show that \(\text{polarisability} \thickapprox \frac{p^2}{3 kT}\) for small \(x\), achieving an accuracy of about one part in \(\frac{15}{x^2}\).

Polarizability

Polarizability measures how much a molecule like water can align its dipole moment with an external electric field. In simpler terms, it tells us how easily a molecule's electron cloud can be distorted by an electric field.
In this exercise, polarizability \(\text{α}\) is derived using the Boltzmann distribution and series expansion of hyperbolic functions. The formula \(\frac{p}{E} (\text{coth} x - x^{-1})\), where \(x = \frac{pE}{kT}\), describes the average polarizability. We find that for small values of \(x\), \(\text{α}\) simplifies to \(\frac{p^2}{3 kT}\). Our accuracy analysis shows that this approximation is reliable, within one part in \(\frac{15}{x^2}\).

Thermal Agitation

Thermal agitation refers to random motion of molecules due to thermal energy. This random movement hinders perfect alignment of dipoles in an external electric field.
The temperature \(T\) causes thermal agitation and directly impacts polarizability. Higher temperatures increase molecular motion, reducing alignment coherence. In the exercise, thermal agitation is accounted for using the Boltzmann constant \(k\) and temperature \(T\). This adjustment explains why dipoles never fully align with the field. This relationship is key to understanding why we ultimately use \(\frac{p^2}{3kT}\) as our simplified polarizability formula, accurately reflecting the influence of both the electric field and thermal agitation.