Thermal Equilibrium

Thermal Equilibrium: Temperature

To understand temperature, we have to look at behavior on the molecular scale. Temperature is essentially a measurement of the average amount of kinetic energy the molecules in an object have. For a given substance, the more kinetic energy the molecules have, the hotter that substance will be. These motions are typically depicted as being vibrations, however, vibration is just one part of it. General back and forth, left and right movement can occur in molecules, as well as rotation. A combination of all of these motions results in a completely random movement of molecules. As well as this, different molecules will move at different rates, and whether or not the state of the matter is a solid, liquid, or gas is also a factor. When a molecule is engaging in this motion, the surrounding molecules are doing the same. As a result of this, many molecules will interact or collide and bounce off of each other. In doing this, molecules will transfer energy between each other, with one gaining energy and one losing it.

An example of a water molecule engaging in random motion due to kinetic energy.

Wikimedia Commons

What Occurs at Thermal Equilibrium?

Now imagine this transfer of kinetic energy occurring between two molecules in two different objects, instead of two in the same object. The object at the lower temperature will have molecules with less kinetic energy, while the molecules in the object at a higher temperature will have more kinetic energy. When the objects are in thermal contact and the molecules can interact, the molecules with less kinetic energy will gain more and more kinetic energy, and in turn, pass that on to the other molecules in the object with a lower temperature. Over time, this carries on until there is an equal value of average kinetic energy in the molecules of both objects, making it so both objects are of an equal temperature - thus achieving thermal equilibrium.

Thermal Equilibrium Formula

When it comes to the transfer of heat energy, it's important to not fall into the trap of using temperature when the calculation is involved. Instead, the word energy is more appropriate, and therefore joules is the better unit. To determine the temperature of equilibrium between two objects of varying temperatures (hot and cold), we must first note that this equation is correct:

\[q_{hot}+q_{cold}=0\]

This equation tells us that the heat energy \(q_{hot}\) lost by the hotter object is the same magnitude but an opposite sign of the heat energy gained by the colder object \(q_{cold}\), measured in joules \(J\). Therefore, adding these two together is equal to 0.

Now, we can calculate the heat energy for both of these in terms of the object properties. To do so, we need this equation:

\[q=m\cdot c\cdot \Delta T\]

Where \(m\) is the mass of the object or substance, measured in kilograms \(kg\), \(\Delta T\) is the temperature change, measured in degrees Celcius \(^{\circ}C\) (or Kelvin \(^{\circ}K\), as their magnitudes are equal) and \(c\) is the specific heat capacity of the object, measured in joules per kilogram Celcius \(\frac{J}{kg^{\circ}C}\).

The only thing we have left to determine here is the temperature change \(\Delta T\) . As we're looking for the temperature at thermal equilibrium, the temperature change can be thought of as the difference between the equilibrium temperature \(T_{e}\) and the current temperatures of each object \(T_{h_{c}}\) and \(T_{c_{c}}\). With the current temperatures being known, and the equilibrium temperature being the variable that we are solving for, we can assemble this rather large equation:

\[m_{h}c_{h}(T_{e}-T_{h_{c}})+m_{c}c_{c}(T_{e}-T_{c_{c}})=0\]

Where anything underscored with an \(h\) regards the hotter object, and anything underscored with a \(c\) regards the colder object. You may notice that we have the variable \(T_{e}\) marked twice in the equation. Once all the other variables are put into the formula, you will be able to combine these into one, to find the final temperature of thermal equilibrium, measured in Celsius.

Thermal Equilibrium Examples

Examples of thermal equilibrium are all around us, and we utilize this phenomenon much more than you may realize. When you're ill, your body might heat up with a fever, but how do we know what temperature it is? We use a thermometer, which uses thermal equilibrium to work. You must have your body in contact with the thermometer for a while, and this is as we have to wait for you and the thermometer to reach thermal equilibrium. Once this is the case, we can deduce that you are at the same temperature as the thermometer. From there, the thermometer simply uses a sensor to determine its temperature at that time, and displays it, in the process showing your temperature as well.

A thermometer uses thermal equilibrium to measure temperature. Wikimedia Commons

Any change of state is also a result of thermal equilibrium. Take an ice cube on a hot day. The hot air is at a much higher temperature than the ice cube, which will be below \(0^{\circ}C\). Due to the large difference in temperature, and the abundance of heat energy in the hot air, the ice cube will eventually melt and reach the temperature of this air over time, with the air only decreasing in temperature by a tiny amount. Depending on how hot the air is, the melted ice may even reach evaporation levels and turn into a gas!

A time-lapse of ice cubes melting due to thermal equilibrium.Wikimedia Commons