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Phase Transitions

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.

Phase transitions: Level 2-4 Challenges

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

Assume that the pot never runs out of water.

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.

Suppose you have a glass of water that contains an ice cube. What happens to the water level in the glass when the ice melts?

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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