KAHIBARO
Table of Contents
What specific heat capacity means
Specific heat capacity tells us how much heat energy is needed to change the temperature of a substance. Different materials warm up at different rates, even if they have the same mass and receive the same amount of energy. For example, water heats up more slowly than many metals. This means water has a larger specific heat capacity.
The specific heat capacity of a substance is defined as the heat required to raise the temperature of unit mass of that substance by one degree.
If a mass $m$ of a substance changes temperature by $\Delta T$ and absorbs or releases heat $Q$, then
$$
Q = mc\Delta T
$$
where $c$ is the specific heat capacity.
In SI units, specific heat capacity is measured in
$$
\mathrm{J \, kg^{-1} \, K^{-1}}
$$
A change of $1 \, \mathrm{K}$ is the same size as a change of $1 \, ^\circ\mathrm{C}$, so temperature differences may be written in either unit.
Important formula:
$$
Q = mc\Delta T
$$
where $Q$ is heat transferred, $m$ is mass, $c$ is specific heat capacity, and $\Delta T = T_f - T_i$ is the temperature change.
Interpreting the formula
The equation shows three simple ideas. If the mass is larger, more energy is needed for the same temperature change. If the temperature change is larger, more energy is needed. If the specific heat capacity is larger, the substance is harder to heat.
A large value of $c$ means the substance can absorb a lot of energy without a large rise in temperature. A small value of $c$ means the temperature changes more easily.
If the substance is heated, then $\Delta T > 0$ and $Q > 0$. If the substance cools, then $\Delta T < 0$ and $Q < 0$.
Sign rule:
If temperature rises, $\Delta T$ is positive and the substance gains heat.
If temperature falls, $\Delta T$ is negative and the substance loses heat.
Everyday meaning
Specific heat capacity helps explain many common observations. Sand on a beach becomes hot quickly in sunlight, but seawater warms more slowly. A metal spoon in hot soup quickly changes temperature, while the soup itself stores much more thermal energy. Buildings near large bodies of water often experience smaller temperature changes because water can absorb and release large amounts of heat.
This idea is especially important in climate, cooking, engineering, and material choice.
Units and dimensions
From the equation
$$
c = \frac{Q}{m\Delta T}
$$
the unit of specific heat capacity is found by dividing energy by mass and temperature change:
$$
\frac{\mathrm{J}}{\mathrm{kg}\cdot \mathrm{K}} = \mathrm{J \, kg^{-1} \, K^{-1}}
$$
Its dimensions are
$$
[c] = \frac{ML^2T^{-2}}{M\Theta} = L^2T^{-2}\Theta^{-1}
$$
where $\Theta$ represents temperature dimension.
Typical values
Different substances have different specific heat capacities. Here are some typical values.
| Substance | Specific heat capacity $c$ $\left(\mathrm{J \, kg^{-1} \, K^{-1}}\right)$ |
|---|---|
| Water | 4186 |
| Ice | 2100 |
| Aluminum | 900 |
| Copper | 385 |
| Iron | 450 |
| Lead | 128 |
| Air | about 1000 |
Water has a particularly high specific heat capacity compared with many common materials. That is why it is useful for cooling systems and for storing thermal energy.
How to use the equation
Suppose $2.0 \, \mathrm{kg}$ of water is heated by $5.0 \, ^\circ\mathrm{C}$. Taking $c = 4186 \, \mathrm{J \, kg^{-1} \, K^{-1}}$, the heat absorbed is
$$
Q = mc\Delta T
$$
$$
Q = (2.0)(4186)(5.0)
$$
$$
Q = 41860 \, \mathrm{J}
$$
So the water absorbs about $4.19 \times 10^4 \, \mathrm{J}$.
Now consider $2.0 \, \mathrm{kg}$ of aluminum heated by the same temperature change, using $c = 900 \, \mathrm{J \, kg^{-1} \, K^{-1}}$:
$$
Q = (2.0)(900)(5.0) = 9000 \, \mathrm{J}
$$
The same mass and same temperature rise require much less energy than for water.
Rearranging the formula
Sometimes you know the heat added and want the temperature change. Rearranging gives
$$
\Delta T = \frac{Q}{mc}
$$
Or, if you want to determine the specific heat capacity experimentally,
$$
c = \frac{Q}{m\Delta T}
$$
These forms are all versions of the same relationship.
Useful rearrangements:
$$
Q = mc\Delta T
$$
$$
\Delta T = \frac{Q}{mc}
$$
$$
c = \frac{Q}{m\Delta T}
$$
Heating and cooling
The same equation works whether energy is added or removed, as long as signs are used consistently. If a hot object cools from temperature $T_i$ to $T_f$, then
$$
\Delta T = T_f - T_i
$$
which is negative. Then $Q$ is also negative, showing that heat left the object.
For example, if $0.50 \, \mathrm{kg}$ of copper cools by $20 \, \mathrm{K}$, then
$$
Q = mc\Delta T = (0.50)(385)(-20)
$$
$$
Q = -3850 \, \mathrm{J}
$$
The negative sign means the copper lost $3850 \, \mathrm{J}$ of heat.
Why substances have different values
Specific heat capacity depends on how a material stores internal energy. When heat is added, the energy does not always go only into increasing the motion associated with temperature. In many substances, energy is shared among different microscopic motions of atoms and molecules. Because of this, some substances need more energy than others for the same temperature increase.
At a beginner level, the key point is simple: the internal structure of a material affects how much energy is needed to raise its temperature.
Constant pressure and constant volume
For gases, the specific heat capacity can depend on how the gas is heated. A gas heated at constant pressure may have a different specific heat capacity from the same gas heated at constant volume. These are often written as $c_p$ and $c_v$.
For solids and liquids in many everyday situations, one value of $c$ is usually enough. For gases, more care is needed because expansion can matter.
For gases, specific heat capacity may depend on the process.
At constant pressure, use $c_p$.
At constant volume, use $c_v$.
A simple comparison diagram
The steeper line represents a smaller specific heat capacity, because the temperature rises more for the same added heat. The flatter line represents a larger specific heat capacity.
Common mistakes
A very common mistake is confusing heat capacity with specific heat capacity. Heat capacity refers to a whole object, while specific heat capacity refers to the material per unit mass. If an object has mass $m$ and material specific heat capacity $c$, then its heat capacity $C$ is
$$
C = mc
$$
Another common mistake is using the final temperature instead of the temperature change. In the equation $Q = mc\Delta T$, you must use
$$
\Delta T = T_f - T_i
$$
not just $T_f$.
Students also sometimes worry about using Celsius or kelvin. For temperature differences, both give the same numerical result.
Do not confuse
$$
T
$$
with
$$
\Delta T
$$
In heating calculations, always use the change in temperature:
$$
\Delta T = T_f - T_i
$$
Summary
Specific heat capacity measures how much heat energy is needed to change the temperature of a unit mass of a substance by one degree. The basic relation is
$$
Q = mc\Delta T
$$
A larger $c$ means the substance resists temperature change more strongly. Water has a high specific heat capacity, which makes it especially important in nature and technology. Understanding this quantity helps connect heat transfer to measurable temperature changes.