Heat Capacity
Imagine a flame placed below a beaker of water. The water warms up and eventually starts to boil. Why? The flame is caused by the combustion of methane (natural gas), which is exothermic. The energy released by the combustion reaction causes molecules in air (mostly N2 and O2) to translate, rotate, and vibrate faster. These high kinetic energy molecules bang into the thin glass beaker, which is composed mostly of silicon and oxygen atoms. These Si and O atoms start to vibrate more because of collisions with the hot air. Next, the water molecules in contact with the beaker warm up because they are in contact with the high kinetic energy Si and O atoms. The water molecules then translate and rotate and vibrate faster too, and they transfer their energy to water molecules throughout the beaker. How much more energy do these water molecules have? Remember that temperature is a measure of the average kinetic energy per molecule. The more energy that is added to the water, the higher the average kinetic energy per molecule, and therefore the higher the temperature. The energy transfer is the form of heat, q, and the temperature change is ΔT. The ratio of these numbers is called the heat capacity, C:
C = 
Think of heat capacity as a measure of the ability of molecules to store energy. Substances with a higher heat capacity have more ways to store energy (energy can be stored as vibrations, rotations, and even hydrogen bonds). The heat capacity can also be higher just because there is more substance to heat up—heat capacity is another extensive property. Therefore, C varies both with the identity and the amount of substance. We can easily measure the heat capacity and then covert it into a molar quantity, Cm, to provide the heat capacity per mole of substance. This gives us tremendous insight into the various ways that atoms and molecules move and store energy. For example, the molar heat capacity of iron is 25 J/(mol·K), just one-third the value for water of 75 J/(mol·K). Why might water have a molar heat capacity so much higher than iron?
Next, let’s looks at some examples and how heat capacity enters our daily lives. For example, consider the heat capacities of two cast iron frying pans. The heat capacity of the large pan is five times greater than that of the small pan because, although both are made of the same material, the mass of the large pan is five times greater than the mass of the small pan. More mass means more atoms are present in the larger pan, so it takes more energy to make all of those atoms vibrate faster.
The heat capacity of the small cast iron frying pan is found by observing that it takes 18,150 J of energy to raise the temperature of the pan by 50.0 °C:
Csmall pan = 
= 363 J/°C
The larger cast iron frying pan, while made of the same substance, requires 90,700 J of energy to raise its temperature by 50.0 °C. The larger pan has a (proportionally) larger heat capacity because the larger amount of material requires a (proportionally) larger amount of energy to yield the same temperature change:
Clarge pan = 
= 1814 J/°C
The specific heat capacity (c) of a substance, commonly called its “specific heat,” is the quantity of heat required to raise the temperature of 1 gram of a substance by 1 degree Celsius (or 1 kelvin):
c = 
Specific heat capacity depends only on the kind of substance absorbing or releasing heat. It is an intensive property—the type, but not the amount, of the substance is all that matters. For example, the small cast iron frying pan has a mass of 808 g. The specific heat of iron (the material used to make the pan) is therefore:
ciron = 
= 0.449 
The large frying pan has a mass of 4040 g. Using the data for this pan, we can also calculate the specific heat of iron:
ciron = 
= 0.449 
Although the large pan is more massive than the small pan, since both are made of the same material, they both yield the same value for specific heat (for the material of construction, iron). Note that specific heat is measured in units of energy per temperature per mass and is an intensive property, being derived from a ratio of two extensive properties (heat and mass). The molar heat capacity, also an intensive property, is the heat capacity per mole of a particular substance and has units of J/(mol·°C).
Liquid water has a relatively high specific heat (about 4.184 J/(g·°C)); most metals have much lower specific heats (usually less than 1 J/(g·°C)). The specific heat of a substance varies somewhat with temperature. However, this variation is usually small enough that we will treat specific heat as constant over the range of temperatures that will be considered in this chapter. Specific heats of some common substances are listed in Table 1. Note that the units of specific heat J/(g·°C), are identical to J/(g·K) and no conversion is needed between the two. This is because the temperature component in the definition of specific heat is a change in temperature, and with 1 °C being exactly the same size as 1 K, it is the same whether expressed in °C or K.
| Substance | Symbol (state) | Specific Heat (J/(g·°C)) |
|---|---|---|
| helium | He(g) | 5.193 |
| water | H2O(ℓ) | 4.184 |
| ethanol | C2H6O(ℓ) | 2.376 |
| ice | H2O(s) | 2.093 (at −10 °C) |
| water vapor | H2O(g) | 1.864 |
| nitrogen | N2(g) | 1.040 |
| oxygen | O2(g) | 0.918 |
| aluminum | Al(s) | 0.897 |
| carbon dioxide | CO2(g) | 0.853 |
| argon | Ar(g) | 0.522 |
| iron | Fe(s) | 0.449 |
| copper | Cu(s) | 0.385 |
| lead | Pb(s) | 0.130 |
| gold | Au(s) | 0.129 |
| silicon | Si(s) | 0.712 |
If we know the mass of a substance and its specific heat, we can determine the amount of heat, q, entering or leaving the substance by measuring the temperature change before and after the heat is gained or lost:
q = (specific heat) × (mass of substance) × (temperature change)
q = c × m × ∆T = c × m × (Tfinal – Tinitial)
In this equation, c is the specific heat of the substance, m is its mass, and ΔT (which is read “delta T”) is the temperature change, Tfinal − Tinitial. If a substance gains thermal energy, its temperature increases (i.e., its final temperature is higher than its initial temperature and Tfinal − Tinitial has a positive value), and the value of q is positive. If a substance loses thermal energy, its temperature decreases (i.e., the final temperature is lower than the initial temperature and Tfinal − Tinitial has a negative value), and the value of q is negative.
Example 1
A flask containing 8.0 × 102 g of water is heated, and the temperature of the water increases from 21 °C to 85 °C. How much heat did the water absorb?
Solution
To answer this question, consider these factors:
- the specific heat of the substance being heated (in this case, water)
- the amount of substance being heated (in this case, 800 g)
- the magnitude of the temperature change (in this case, from 21 °C to 85 °C).
The specific heat of water is 4.184 J/(g·°C), so to heat 1 g of water by 1 °C requires 4.184 J. We note that since 4.184 J is required to heat 1 g of water by 1 °C, we will need 800 times as much to heat 800 g of water by 1 °C. Finally, we observe that since 4.184 J are required to heat 1 g of water by 1 °C, we will need 64 times as much to heat it by 64 °C (that is, from 21 °C to 85 °C).
This can be summarized using the equation:
q = c × m × ∆T = c × m × (Tfinal – Tinitial) = (4.184 ) × (800 g) × (85 °C – 21 °C)
q = (4.184 ) × (800 g) × (64 °C) = 210,000 J = 210 kJ
Because the temperature increased, the water absorbed heat and q is positive.
Check Your Learning
How much heat, in joules, must be added to a 5.00 × 102 g iron skillet to increase its temperature from 25 °C to 250 °C? The specific heat of iron is 0.449 J/(g·°C).
Answer
5.05 × 104 J (50.5 kJ)
Note that the relationship between heat, specific heat, mass, and temperature change can be used to determine any of these quantities (not just heat) if the other three are known or can be deduced.
Example 2
A piece of unknown metal weighs 348 g. When the metal piece absorbs 6.64 kJ of heat, its temperature increases from 22.4 °C to 43.6 °C. Determine the specific heat of this metal (which might provide a clue to its identity).
Solution
Since mass, heat, and temperature change are known for this metal, we can determine its specific heat using the relationship:
q = c × m × ∆T = c × m × (Tfinal – Tinitial)
Substituting the known values:
6640 J = c × (348 g) × (43.6 °C – 22.4 °C)
Solving:
c = 
= 0.900 J/(g·°C)
Comparing this value with the values in Table 1, this value matches the specific heat of aluminum, which suggests that the unknown metal may be aluminum. Note that we would need more information to make a definitive determination of the identity.
Check Your Learning
A piece of unknown metal weighs 217 g. When the metal piece absorbs 1.43 kJ of heat, its temperature increases from 24.5 °C to 39.1 °C. Determine the specific heat of this metal, and predict its identity (use Table 1 for reference).
Answer
c = 0.451 J/(g·°C); the metal is likely to be iron
Chemistry in Real Life: Solar Thermal Energy Power Plants
The sunlight that reaches the earth contains thousands of times more energy than we presently capture. Solar thermal systems provide one possible solution to the problem of converting energy from the sun into energy we can use. Large-scale solar thermal plants have different design specifics, but all concentrate sunlight to heat some substance. The heat “stored” in that substance is then converted into electricity.
The Solana Generating Station in Arizona’s Sonora Desert produces 280 megawatts of electrical power. It uses parabolic mirrors that focus sunlight on pipes filled with a heat transfer fluid (HTF) (Figure 1). The HTF then does two things: It turns water into steam, which spins turbines, which in turn produces electricity, and it melts and heats a mixture of salts, which functions as a thermal energy storage system. After the sun goes down, the molten salt mixture can then release enough of its stored heat to produce steam to run the turbines for 6 hours. Molten salts are used because they possess a number of beneficial properties, including high heat capacities and thermal conductivities.
The 377-megawatt Ivanpah Solar Generating System, located in the Mojave Desert in California, is the largest solar thermal power plant in the world (Figure 2). Its 170,000 mirrors focus huge amounts of sunlight on three water-filled towers, producing steam at over 538 °C that drives electricity-producing turbines. It produces enough energy to power 140,000 homes. Water is used as the working fluid because of its large heat capacity and heat of vaporization.
