Question

Piezoelectricity. Apply a mechanical force \(f\) along the \(x\) axis of a piezoelectric crystal, such as quartz, which has dimension \(\ell\) in that direction. The crystal will develop a polarization \(p_{0}\), a separation of charge along the \(x\) axis, positive on one face and negative on the other. Applying an electric field \(E\) along the \(x\) axis causes a mechanical deformation. Such devices are used in microphones, speakers, and pressure transducers. For such systems, the energy equation is \(d U=T d S+f d \ell+E d p_{0}\). Find a Maxwell relation to determine \(\left(\partial p_{0} / \partial f\right)_{T, E}\).

Short answer

The Maxwell relation is \( \left( \frac{\partial p_{0}}{\partial f} \right)_{T, E} = \left( \frac{\partial \ell}{\partial E} \right)_{T, f} \).

Step-by-step solution

- Identify the energy equation

The energy equation is given as:\[ dU = T dS + f d \, \ell + E d p_{0} \]

- Recognize variables and natural coordinates

In the context of thermodynamics, the energy \( U \) can be expressed as a function of entropy \( S \), dimension along the x-axis \( \ell \), and polarization \( p_{0} \). These are the natural coordinates in this problem.

- Write the total differential of U

The total differential for U considering natural coordinates \( S, \ell, p_{0} \) is:\[ dU = \left( \frac{\partial U}{\partial S} \right)_{\ell, p_{0}} dS + \left( \frac{\partial U}{\partial \ell} \right)_{S, p_{0}} d \, \ell + \left( \frac{\partial U}{\partial p_{0}} \right)_{S, \, \ell} dp_{0} \]

- Match terms with given energy equation

From the given energy equation, match terms to identify partial derivatives:- \( T = \left( \frac{\partial U}{\partial S} \right)_{\ell, p_{0}} \)- \( f = \left( \frac{\partial U}{\partial \ell} \right)_{S, p_{0}} \)- \( E = \left( \frac{\partial U}{\partial p_{0}} \right)_{S, \ell} \)

- Formulate the Maxwell relation

Use the mixed partial derivative property: \[ \frac{\partial}{\partial f} \left( \frac{\partial U}{\partial p_{0}} \right)_{S, \ell} = \frac{\partial}{\partial p_{0}} \left( \frac{\partial U}{\partial f} \right)_{S, \ell} \]

- Express the desired Maxwell relation

From the property and matched terms:\[ \left( \frac{\partial E}{\partial f} \right)_{T, \ell} = \left( \frac{\partial p_{0}}{\partial \ell} \right)_{T, E} \]Recognize that fixing \(T\) and \(E\) implies substituting the corresponding partial derivatives appropriately.

- Final expression for the Maxwell relation

Thus, the Maxwell relation that determines \( \left( \frac{\partial p_{0}}{\partial f} \right)_{T, E} \) is:\[ \left( \frac{\partial p_{0}}{\partial f} \right)_{T, E} = \left( \frac{\partial \ell}{\partial E} \right)_{T, f} \]

Key concepts

Maxwell relations

Maxwell relations are powerful equations in thermodynamics that connect various partial derivatives of thermodynamic potentials. They arise from the equality of mixed second partial derivatives and are instrumental in translating hard-to-measure quantities into more accessible forms.
For instance, consider the energy differential given in the problem: \[ dU = TdS + fd\,\ell + Edp_0 \]By identifying the natural coordinates (\( S, \ell, p_0 \)) and matching terms, we get:

• \( T = \left( \frac{\partial U}{\partial S} \right)_{\ell, p_0} \)
• \( f = \left( \frac{\partial U}{\partial \ell} \right)_{S, p_0} \)
• \( E = \left( \frac{\partial U}{\partial p_0} \right)_{S, \ell} \)

The mixed partial derivative property tells us that:\[ \frac{\partial}{\partial f} \left( \frac{\partial U}{\partial p_0} \right)_{S, \ell} = \frac{\partial}{\partial p_0} \left( \frac{\partial U}{\partial f} \right)_{S, \ell} \]Breaking this down gives us the Maxwell relation:\[ \left( \frac{\partial p_0}{\partial f} \right)_{T,E} = \left( \frac{\partial \ell}{\partial E} \right)_{T,f} \]This relation helps us find the change in polarization with respect to an applied force, using the change in dimension with respect to an electric field.

Thermodynamic equations

Thermodynamic equations describe how different properties within a system relate to each other and change under various conditions. They typically involve variables such as temperature (\( T \)), entropy (\( S \)), pressure (\( P \)), volume (\( V \)), and energy (\( U \)). In this context, the energy differential provided is:\[ dU = TdS + fd\ell + Edp_0 \]This equation shows how the internal energy (\( U \)) changes with small increments of entropy (\( S \)), dimension (\( \ell \)), and polarization (\( p_0 \)).
Each term in this equation corresponds to a different physical process:

• \( TdS \): Change in internal energy due to heat transfer.
• \( fd\ell \): Change due to mechanical work done.
• \( Edp_0 \): Change due to electrical work affecting polarization.

Understanding this helps us see how different inputs (heat, mechanical forces, electric fields) interact to change a system's state. By breaking it down, solving for specific partial derivatives becomes easier and it paves the way for using Maxwell relations effectively.

Energy differential

An energy differential equation is a small change in the total energy of a system represented as the sum of small changes in various variables that make up that system. Consider the given energy differential equation once again:\[ dU = TdS + fd\ell + Edp_0 \]This shows how a small increment in internal energy (\( dU \)) results from changes in entropy, dimension, and polarization. Breaking it down:

• \( dS \): Small change in entropy.
• \( d\ell \): Small change in the dimension along the x-axis.
• \( dp_0 \): Small change in polarization.

Each term's coefficient (like \( T, f, E \)) provides insight into how sensitive the energy is to those changes. Matching these with natural coordinates' expressions yields partial derivatives, fundamental to understanding thermodynamic behaviors. This assists in deriving relations like Maxwell's which connect different thermodynamic quantities, simplifying complex experimental measurements into theoretically manageable forms.