Heat capacity is the ability of a material to absorb heat without directly reflecting all of it as a rise in temperature. You should read the sections on heat and temperature as background, and the water section would help, too.

As heat is added uniformly to like quantities of different substances, their temperatures can rise at different rates. For example, metals,

good conductors of heat, show fast temperature rises when heated. It is relatively easy to heat a metal until it glows red. On the other hand, water can absorb a lot of heat with a relatively small rise in temperature. Insulating materials (insulators) are very poor conductors of heat, and are used to isolate materials that need to be kept at different temperatures — like the inside of your house from the outside.

This graph shows the rise in temperature as heat is added at the same rate to equal masses of aluminium (Al) and water (H2O). The temperature of water rises much more slowly than that of Al.

In the metal, Al atoms only have translational kinetic energy (although that motion is coupled strongly to neighbor atoms). Water, on the other hand, can rotate and vibrate as well. These degrees of freedom of motion can absorb kinetic energy without reflecting it as a rise in temperature of the substance.

aluminum & water T vs. heat added

Equipartition of Energy

Most substances obey the law of equipartition of energy over a broad range of temperatures. The law says that energy tends to be distributed evenly among all of the degrees of freedom of a molecule — translation, rotation and vibration. This has consequences for substances with more or fewer atoms. In the diagram below, each container represents a degree of freedom. The situations for a 3-atom and a 10-atom

 

molecule are shown. If the same total amount of heat energy is added to each molecule, the 3-atom molecule ends up with more energy in its translational degrees of freedom. Because the 10-atom molecule has more vibrational modes in which to store kinetic energy, less is available to go into the translational modes, and it is mostly the translational energy that we measure as temperature.

 

equipartition of energy cartoon

Specific Heat

There's one more refinement left to make to heat capacity. Obviously, the amount of heat required to raise the temperature of a large quantity of a substance is greater than the amount required for a small amount of the same substance. To control for the amount, we generally measure and report heat capacities as specific heat, the heat capacity per unit mass. Specific heats of a great many substances have been measured under a variety of conditions. They are tabulated in books an on-line.

Specific heat is the heat capacity per unit mass.

The specific heat of water is 1 cal/g˚C = 4.184J/g˚C

We generally choose units of J/gram or KJ/Kg. The specific heat of liquid water is 4.184 J/g, which is also 4.184 KJ/Kg. The calorie is a unit of heat defined as the amount of heat required to raise the temperature of 1 cm3 of water by 1˚C.

Calculating Heat and Temperature Changes

The heat, q, required to raise the temperature of a mass, m, of a substance by an amount ΔT is

q = mCΔT = mC(Tf - Ti)

where C is the specific heat and Tf and Ti are the final and initial temperatures. The slope of a graph of temperature vs. heat added to a unit mass is just 1/C.

Using this formula, it's relatively easy to calculate heat added, final or initial temperature or the specific heat itself (that's how it's measured) if the other variables are known.

 

Exercises — Simple heat calculations

(Use the table below to look up missing specific heats.)

  1. How much heat (in Joules) does it take to raise the temperature of 100 g of H2O from 22˚C to 98˚C?
  1. If it takes 640J of heat energy to increase the temperature of 100 g of a substance by 25˚C (without changing its phase), calculate the specific heat of the substance.
  1. If 80 J of heat are added to 100 ml of ethanol (density = 789 Kg·m-3) initially at 10˚C, calculate the final temperature of the sample.

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Exercises — Heat with a phase change

(Use the table below to look up missing specific heats; heats of fusion or vaporization are given.)

  1. How much heat (in Joules) does it take to change 120g of ice at -10˚C to water at 37˚C? (ΔHf= 334 KJ·Kg-1)? Note that this is a three-step problem: First heat the ice to 0˚C, then convert all 120g to liquid, then raise the temperature of the water to 37˚C (human body temp.).
  1. How much heat is released when 1 Kg of steam at 300˚C is cooled to liquid at 40˚C? (ΔHv= 2260 KJ·Kg-1)
  1. Is there enough heat in 100 ml of water at 25˚C to completely melt 50g of ice at 0˚C? (ΔHf= 334 KJ·Kg-1)

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Specific Heats of Selected Elements & Compounds
Compound

Specific Heat

J·g-1·K-1

Compound

Specific Heat

J·g-1·K-1

water (ice)
2.11
methanol (CH3OH)
2.14
water (liquid)
4.184
ethanol (C2H5OH)
2.44
water (steam)
2.08
ethylene glycol (C2O2H6)
2.2
aluminum (s)
0.897
hydrogen (H2) gas
14.267
copper (s)
0.385
benzene (C6H6)
1.750
iron (s)
0.450
wood (typical)
1.674
lead (s)
0.129
glass (typical)
0.867

 

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