How do I calculate the amount of ice needed to cool water to a certain temperature?

	Vocabulary
	calorimetry heat latent heat molar heat of fusion phase change

1. List any important energy sources and sinks. Obviously, the water is the heat source and the ice is the heat sink. The significant energy flows are:
   1. heat absorbed by the ice to warm it to 0°C (call it q₁)
   2. heat absorbed by the ice to melt it at 0°C (call it q₂)
   3. heat absorbed by the melted ice to warm it from 0.0 to 1.0 °C (call it q₃)
   4. heat released by the water to cool it from 25.0°C to 1.0 °C (call it q₄)
   Why is the heat absorbed by the ice broken into three separately considered parts? A phase change (solid to liquid) is involved. Phases changes always absorb or release a certain amount of heat; in this case, heat is supplied to the ice at 0° C to break down the ice lattice and make a less orderly liquid. This heat is sometimes called a "latent heat" because heat is supplied but no temperature change occurs. Melting won't occur until the ice has warmed up to 0°C, so you have to consider the heat required to warm the ice and the heat required to effect the phase change separately. After melting, you'll have water at 0°C, so if you want to warm it to 1°C you have a third amount of heat to include.
2. Set up an energy conservation equation. If all significant transfers of energy were listed in the previous step, the total energy absorbed should be balanced by the total energy released or you'll violate the law of conservation of energy. The best way to keep the signs of the energy changes straight is to simply note that all of the heats should sum to zero. For this problem the equation is:

   q₁ + q₂ + q₃ + q₄ = 0

3. Relate heats to experimental measurements. When an object is simply warming or cooling, you can replace the heats in the conservation equations using q = mcT where m is the mass, c is the specific heat, and T is the temperature change each component undergoes. When a phase change occurs, use q = n H, where n is the number of moles and H is the latent heat for the phase change per mole of substance. For melting, the latent heat per mole is called a "molar heat of fusion" and is given the symbol H. Since

   q₁ = m_ice c_ice T_ice
   q₂ = n_ice H_fus
   q₃ = m_ice c_water T_{melted ice}
   q₄ = m_water c_water T_water
   

   The conservation equation becomes

   m_icec_ice(+10°C) + n_iceH_fus + m_icec_water(+1°C) + m_water c_water(-24°C)  =  0

   where n_ice is moles of ice (which you can easily relate to the mass of ice, right?)
4. Solve the conservation equation. The chemistry is finished. The algebra begins here. You need to get m_ice all by itself on one side of the equation. You also need to look up the specific heats of ice and water, and the molar heat of fusion of water. All of the energy units must be consistent (e. g. if the specific heats are in J/g°C and the molar heat of fusion is in J/mol, everything will work out). Can you take it from here?

Author: Fred Senese senese@antoine.frostburg.edu