Classical Mechanics
# Phase Transitions

A cup of warm water is suspended in a large pot of water held at a steady boil at atmospheric pressure. Will the water in the cup ever boil?

**Assume** that the pot never runs out of water and that the environment remains unchanged.

If the chocolate pope is not to be melt on a hot sunny day, what should he be made of?

You've found a stockpile of 1955 United States pennies, which are now worth approximately twice as much as raw metal compared to their value as currency. A Fresnel lens can be used to concentrate incoming light onto a focal point.

For how many seconds would you need to focus the Fresnel lens on a penny in order to melt the entire coin?

**Details and assumptions:**

- A penny weighs $2.5 \text{ g}$ and is made of $100$% $\text{Cu}.$
- The penny starts out at $25^\circ\text{C}$ and melts at $1085^\circ\text{C}.$
- $\text{Cu}$'s heat of melting is $176 \text{ kJ}/\text{kg}$ and its specific heat is $0.386\text{ kJ}/\text{kg}\cdot\text{K}.$
- The area of the lens is $0.2 \text{ m}^2,$ the power of the sun is $1370\text{ W/m}^2,$ and $100$% of the energy goes into heating the coin.

$\mathcal{F}=(T-T_c)m^2+\frac14 m^4$ where $T$ is the temperature, and $m$ represents some property of the system, such as its overall magnetization. As $T$ rises and falls, but does not cross $T_c$, the allowed physical value(s) of $m$, $m^*$ change somewhat (see Assumptions).

Suppose that the free energy of a system is described byWhen $\displaystyle T$ crosses $T_c$, however, there is a qualitative shift in the allowed values of $m^*$, and the system undergoes a phase transition.

Suppose we're working with the system at a temperature $\displaystyle T$ below $T_c$ such that $T-T_c=-10$, find $\lvert m^*\rvert$.

**Assumptions**

- The system always resides in states $m^*$ which minimize its free energy. These are the "physical" states of the system.

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