Temperature (Physics): Definition, Formula & Examples

You may already have an intuitive sense that temperature is a measure of the "coldness" or "hotness" of an object. Many people are obsessed with checking the forecast so they know what the temperature will be for the day. But what does temperature really mean in physics?

Temperature is a measure of average kinetic energy per molecule in a substance. It is different from heat, although the two quantities are intimately related. Heat is the energy transferred between two objects at different temperatures.

Any physical substance to which you might attribute the property of temperature is made of atoms and molecules. Those atoms and molecules do not stay still, even in a solid. They are constantly moving and jiggling around, but the motion happens on such a small scale, that you can't see it.

As you likely recall from your study of mechanics, objects in motion have a form of energy called ​kinetic energy​ that is associated with both their mass and how fast they are moving. So when temperature is described as average kinetic energy per molecule, it is the energy associated with this molecular motion that is being described.

There are many different scales by which you might measure temperature, but the most common ones are Fahrenheit, Celsius and Kelvin.

The Fahrenheit scale is what those who live in the United States and a few other countries are most familiar with. On this scale water freezes at 32 degrees Fahrenheit, and the temperature of boiling water is 212 F.

The Celsius scale (sometimes also referred to as centigrade) is used in most other countries around the world. On this scale the freezing point of water is at 0 C and the boiling point of water is at 100 C.

The Kelvin scale, named for Lord Kelvin, is the scientific standard. Zero on this scale is at absolute zero, which is where all molecular motion stops. It is considered an absolute temperature scale.

To convert from Celsius to Fahrenheit, use the following relationship:

\(T_F = \frac{9}{5}T_C + 32\)

Where ​T​_​F​ is the temperature in Fahrenheit, and ​_T_C_​ is the temperature in Celsius. For example, 20 degrees Celsius is equivalent to:

\(T_F = \frac{9}{5}20 + 32 = 68\text{ degrees Fahrenheit.}\)

To convert in the other direction, from Fahrenheit to Celsius, use the following:

\(T_C = \frac{5}{9}(T_F – 32)\)

To convert from Celsius to Kelvin, the formula is even simpler because the increment size is the same, and they just have different starting values:

\(T_K=T_C+273.15\)

When two objects at different temperatures are in contact with each other, heat transfer will occur, with heat flowing from the object at the higher temperature to the object at the lower temperature until thermal equilibrium is reached.

This transfer occurs due to collisions between the higher-energy molecules in the hot object with the lower-energy molecules in the cooler object, transferring energy to them in the process until enough random collisions between molecules in the materials have occurred that the energy becomes equally distributed between the objects or substances. As a result, a new final temperature is achieved, which lies between the original temperatures of the hot and the cool objects.

Another way to think of this is that the total energy contained in both substances eventually becomes equally distributed between the substances.

The final temperature of two objects at different initial temperatures once they reach thermal equilibrium can be found by using the relationship between heat energy ​Q​, specific heat capacity ​c​, mass ​m​ and the temperature change given by the following equation:

\(Q = mc\Delta T\)

​Example:​ Suppose 0.1 kg of copper pennies (​_c_c​ = 390 J/kgK) at 50 degrees Celsius are dropped into 0.1 kg of water (​c_w_​ = 4,186 J/kgK) at 20 degrees Celsius. What will the final temperature be once thermal equilibrium is achieved?

Solution: Consider that the heat added to the water from the pennies will equal the heat removed from the pennies. So if the water absorbs heat ​_Q_w_​ where:

\(Q_w = m_wc_w\Delta T_w\)

Then for the copper pennies:

\(Q_c=-Q_w = m_cc_c\Delta T_c\)

This allows you to write the relationship:

\(m_cc_c\Delta T_c=-m_wc_w\Delta T_w\)

Then you can make use of the fact that both the copper pennies and the water should have the same final temperature, ​_T_f_​, such that:

\(\Delta T_c=T_f-T_{ic}\Delta T_w=T_f-T_{iw}\)

Plugging these ​ΔT​ expressions into the previous equation, you can then solve for ​_T_f_​. A little algebra gives the following result:

\(T_f = \frac{m_cc_c T_{ic}+m_wc_w T_{iw}}{m_cc_c+m_wc_w}\)

Plugging in the values then gives:

\(T_f = \frac{0.1\times 390\times 50+0.1\times 4186\times 20}{0.1\times 390+0.1\times 4186}= 22.6 \text{ degrees Celsius}\)

Note: If you're surprised that the value is so close to the water's initial temperature, consider the significant differences between the specific heat of water and the specific heat of copper. It takes a lot more energy to cause a temperature change in water than it does to cause a temperature change in copper.

Old-fashioned glass-bulb mercury thermometers measure temperature by making use of the thermal expansion properties of mercury. Mercury expands when warm and contracts when cool (and to a much larger degree than the glass thermometer which contains it does.) So as the mercury expands, it rises inside the glass tube, allowing for measurement.

Spring thermometers – those that usually have a circular face with a metal pointer – also work off of the principle of thermal expansion. They contain a piece of coiled metal that expands and cools based on temperature, causing the pointer to move.

Digital thermometers make use of heat-sensitive liquid crystals to trigger digital temperature displays.

While temperature is a measure of the average kinetic energy per molecule, internal energy is the total of all of the kinetic and potential energies of the molecules. For an ideal gas, where potential energy of the particles due to interactions is negligible, the total internal energy ​E​ is given by the formula:

\(E = \frac{3}{2}nRT\)

Where ​n​ is the number of moles and ​R​ is the universal gas constant = 8.3145 J/molK.

Not surprisingly, as temperature increases, thermal energy increases. This relationship also makes it clear why the Kelvin scale is important. The internal energy should be any value 0 or greater. It would never make sense for it to be negative. Not using the Kelvin scale would complicate the internal energy equation and require the addition of a constant to correct it. The internal energy becomes 0 at absolute 0 K.