Classical Mechanics

Phase Transitions

Phase transitions: Level 2-4 Challenges


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 g2.5 \text{ g} and is made of 100100% Cu.\text{Cu}.
  • The penny starts out at 25C25^\circ\text{C} and melts at 1085C.1085^\circ\text{C}.
  • Cu\text{Cu}'s heat of melting is 176 kJ/kg176 \text{ kJ}/\text{kg} and its specific heat is 0.386 kJ/kgK.0.386\text{ kJ}/\text{kg}\cdot\text{K}.
  • The area of the lens is 0.2 m2,0.2 \text{ m}^2, the power of the sun is 1370 W/m2,1370\text{ W/m}^2, and 100100% 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 F=(TTc)m2+14m4\mathcal{F}=(T-T_c)m^2+\frac14 m^4 where TT is the temperature, and mm represents some property of the system, such as its overall magnetization. As TT rises and falls, but does not cross TcT_c, the allowed physical value(s) of mm, mm^* change somewhat (see Assumptions).

When T\displaystyle T crosses TcT_c, however, there is a qualitative shift in the allowed values of mm^*, and the system undergoes a phase transition.

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


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

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