Lecture 9: Rods, bubbles, and magnets

Note

So I could ask myself really what is goin’ wrong? – Black Eyed Peas, Where is the love?

Warning

This lecture corresponds to Chapter 17 of the textbook.

Summary

Attention

In this chapter, we examine various examples of work situations, beyond the $\dbar W=-pdV$ that has been front an center so far, as we described the reversible work done on a gas held at constant pressure $p$ . Here, instead, we use the machinery developed in Lecture 9: Rods, bubbles, and magnets to obtain information on thermodynamical properties of: rods, bubbles, and magnets.

The functional forms of the change in work added (positive) to or taken away (negative) from a system are summarized in this table:

$\begin{array}{llll} \hline & X & x & d W \\ \hline \text { fluid } & -p & V & -p d V \\ \text { elastic rod } & f & L & f d L \\ \text { liquid film } & \gamma & A & \gamma d A \\ \text { dielectric } & E & p _{ E } & - p _{ E } \cdot d E \\ \text { magnetic } & B & m & - m \cdot d B \\ & & & \\ \hline \end{array}$

For a rod, the work is the work related to the elongation of the rod under tension. For a liquid film (e.g., bubble), the work is the energy needed to create a surface $A$ with surface tension $\gamma$ . Finally, we carefully derived an expression for the work done in a system of independent magnets in the presence of an external magnetic field.

We found that depending on how the material of a rod is structured microscopically, we can have an increase or a decrease in entropy upon elongation. For instance, rubber consists of long-chain molecules. With no force applied, the rubber molecule is significantly coiled up and the entropy is large, since there are many different ways to coil up rubber. When a force is applied the molecule becomes more and more linear along the direction of the applied force, and the end-to-end distance is large, thereby reducing the entropy. This reduction in entropy corresponds to heat flowing out of the system. This phenomena is in strong contrast with a metallic rod, which is typically made up of small (highly-ordered) crystallites. When elongated, the crystallites in the rod lose order and entropy increases. These effects are linked to thermal expansion behavior: whereas a metallic rod expands upon heating, a rubber rod coils up when heated up.

We also study in details the work related to forming a liquid surface. After deriving an expression for the pressure inside the bubble, we studied how the entropy changes upon formation of the bubble. When formed isothermally, the internal energy of a bubble increases due to two contributions: the work done to create the surface and the heat needed to keep the temperature constant. The formation of a bubble thus corresponds to an increase of entropy, compared to the bulk.

Finally, this chapter focuses on the very important problem of paramagnetism. In a nutshell, it is the theory that describes the alignment of magnetic dipole with an external magnetic field, so as to reduce the internal energy. The alignment strength is governed by the magnetic susceptibility, which, according to Curie’s law, varies as:

$\chi \propto \frac{1}{T}.$

Students will remember that this theory assumes that we can neglect interactions between magnetic dipoles. We will come back to that problem in the next chapter when we take a deeper dive into how the entropy behaves at very low temperature.

We conclude the chapter with the very important concepts of isothermal magnetization and adiabatic demagnetization (see blue box below).

Note

Isothermal magnetization and adiabatic demagnetization

Note that all the processes described here are considered reversible. Using one of Maxwell’s relations, we find that the isothermal magnetization is given by:

$\left(\frac{\partial S}{\partial B}\right)_{T}=\left(\frac{\partial m}{\partial T}\right)_{B} \approx \frac{V B}{\mu_{0}}\left(\frac{\partial \chi}{\partial T}\right)_{B}.$

The heat absorbed in the isothermal change of $B$ is

$\Delta Q=T\left(\frac{\partial S}{\partial B}\right)_{T} \Delta B=\frac{T V B}{\mu_{0}}\left(\frac{\partial \chi}{\partial T}\right)_{B} \Delta B<0.$

The negative sign is due to Curie’s law (see above). It indicates that heat is released from the system: the entropy decreases! This is not surprising since the magnetic moments start to align and there are fewer and fewer microstates possible as the alignment evolves.

Once magnetized, the system can be demagnetized. If we imagine the demagnetization is done adiabatically, we see that:

$\left(\frac{\partial T}{\partial B}\right)_{S}=-\frac{T V B}{\mu_{0} C_{B}}\left(\frac{\partial \chi}{\partial T}\right)_{B}>0.$

Therefore, we see that the temperature goes down as $B$ goes down. We can understand why the temperature goes down as follows: at high magnetic field, all moments are aligned (low entropy). During the adiabatic demagnetization, the change in entropy is zero (remember that the process described here is reversible, thus we can use this equation $dS=\dbar Q_{\rm rev}/T=0$ ). However, as the magnetic moment goes down, the entropy due to the alignment of the moments goes up! To ensure isentropicity of the system, there is a reduction of entropy of the other degrees of freedom of the system. Namely, the elementary vibrations of the system decrease their entropy, which leads to a cooling effect.

Key Definitions

Note

Isothermal Young’s modulus:: ratio of stress $\sigma$ to strain $\epsilon$ :

$E_{T}=\frac{\sigma}{\epsilon}=\frac{L}{A}\left(\frac{\partial f}{\partial L}\right)_{T}.$
Linear expansivity at constant tension:: $\alpha_{f}=\frac{1}{L}\left(\frac{\partial L}{\partial T}\right)_{f}.$
Surface tension:: energy required to create a unit of surface of a liquid.
Paramagnetism:: physical effect that tends to have magnetic moments aligned parallel to an applied magnetic field. The strength of the alignment is governed by the magnetic susceptibility. This effect is described by Curie’s law.

A full list of terms, including the ones provided here, can be found in the Index.

Learning Material

Copy of Slides

The slides for Lecture 9 are available in pdf format here: pdf

Screencast

Test your knowledge

Imagine a one-dimensional rod of a given material. How does the entropy change when you stretch it isothermally?
1. It remains constant, regardless of how you stretch it.
2. It always goes up and this is accompanied by a release of heat.
3. It always goes down and this is accompanied by an uptake of heat.
4. The answer to this question depends on what material the rod is made of.
Imagine a droplet of water with surface tension $\gamma$ and surface area $A$ .
1. The quantity $\dbar W=\gamma \textrm{d}A$ is the work needed to increase the surface area from $A$ to $A+\textrm{d}A$ .
2. The quantity $\dbar W=-\gamma \textrm{d}A$ is the work needed to increase the surface area from $A$ to $A+\textrm{d}A$ .
3. The quantity $\dbar Q=\gamma \textrm{d}A$ is the heat you need to transfer to the droplet to evaporate it.
The pressure in a bubble is always…
1. The same as the pressure outside of it.
2. Larger than the outside pressure, with a difference proportional to the surface tension.
3. Smaller than the outside pressure, with a difference proportional to the surface tension.
Curie law for a paramagnetic substance assumes…
1. magnetic moments do not interact among themselves or with an external magnetic field
2. magnetic moments do not interact among themselves, but they do interact with an external magnetic field
3. magnetic moments interact among themselves but not with an external magnetic field
4. magnetic moments interact among themselves and with an external magnetic field
What is paramagnetism?
1. A phenomenon where magnetic moments align with an external field.
2. A magnet that safely drops from an airplane.
3. A synonym of ferromagnetism.

Hint

Find the answer keys on this page: Answers to selected test your knowledge questions. Don’t cheat! Try solving the problems on your own first!

Homework Assignment

Solve the following problems from the textbook: