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A sparsely planted forest is likely to have small scale fires while a densely packed one has the potential to burn completely — this shift is called a phase transition.

A cup of warm water is suspended in a large pot of boiling water. Will the water in the cup ever boil?

Assume that the pot never runs out of water.

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

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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.

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Suppose that the free energy of a system is described by \[\mathcal{F}=(T-T_c)m^2+\frac14 m^4\] 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).

When \(\displaystyle T\) 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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