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How do I calculate the amount of ice needed to cool water to a certain temperature?
- You have a supply of ice in a freezer at -10.0 C and a glass containing 150 g of water at 25.0 C. How many grams of ice at -10.0 C must be added to the water to reduce the temperature of the water to 1.0 C? Ignore the heat capacity of the glass container.
To solve a problem that ties temperature changes to heat
flows, you'll have to set up and solve a conservation equation that relates changes in the
energies of the objects you're interested in. Follow these steps:
- 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:
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.
- heat absorbed by the ice to warm it to 0°C (call it q1)
- heat absorbed by the ice to melt it at 0°C (call it q2)
- heat absorbed by the melted ice to warm it from 0.0 to 1.0 °C (call it q3)
- heat released by the water to cool it from 25.0°C to 1.0 °C (call it q4)
- 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
- 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 = mice cice Tice
The conservation equation becomes
q2 = nice Hfus
q3 = mice cwater Tmelted ice
q4 = mwater cwater Twater
micecice(+10°C) + niceHfus + micecwater(+1°C)
+ mwater cwater(-24°C) = 0
where nice is moles of ice (which you can easily relate to the mass of ice, right?)
- Solve the conservation equation. The chemistry is finished. The algebra
begins here. You need to get mice 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 firstname.lastname@example.org