In which case two bodies are described to be in thermal equilibrium

Like it or not, thermal equilibrium is a big part of our lives. We naturally expect cold things to eventually get warmer, and we plan for hot things eventually cooling down, reaching an equilibrium of temperature. Thermal equilibrium is something that happens to us and something that we use, but it may not be obvious to us. Given long enough, thermal equilibrium is theoretically eventually reached whenever two objects or substances of different temperatures are in contact. But what is thermal equilibrium, how do we calculate it, and where is it used in everyday life? Let's find out.

Thermal Equilibrium Definition

Thermal equilibrium occurs when two or more objects or thermodynamic systems are connected in a way where energy can transfer (also known as thermal contact), and yet there is no net flow of heat energy between the both of them.

A thermodynamic system is a defined region of space with theoretical walls that separate it from the surrounding space. The permeability of these walls to energy or matter depends on the type of system.

This typically means no heat energy flows between them, but this can also mean that as energy flows into one system from the other, that system will also transfer the same amount of energy right back, making the net amount of heat transferred 0.

Thermal equilibrium is heavily related to the field of thermodynamics and its laws. Specifically, the zeroth law of thermodynamics.

The zeroth law of thermodynamics states that: if two thermodynamic systems are each separately in thermal equilibrium with a third system, then they are also in thermal equilibrium with one another.

When thermal equilibrium is reached, both objects or systems are at the same temperatures, with no net transfer of heat energy taking place between them.

Thermal equilibrium can also mean an even distribution of thermal energy throughout a single object or body. Thermal energy in a single system does not immediately have an equal level of heat across its entirety. If an object is heated, the point on the object or system at which thermal energy is applied will initially be the area with the highest temperature whereas other regions on or in the system will have a lower temperature. The initial distribution of heat in the object will depend on a range of factors including the material properties, geometry, and how the heat was applied. However, over time the heat energy will disperse throughout the system or object, eventually reaching an internal 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.

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.

One of the underlying reasons that objects or systems in thermal contact will eventually reach thermal equilibrium is the second law of thermodynamics. The second law states that energy in the universe is constantly moving towards a more disordered state by increasing the amount of entropy.

A system containing two objects is more ordered if one object is hot and one cold, therefore the entropy is increased if both objects become the same temperature. This is what drives heat to transfer between objects of different temperatures until thermal equilibrium is reached, which represents the state of maximum entropy.

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 \(\frac{J}{kg^{\circ}C}\).

Specific heat capacity is a material property, meaning it is different depending on the material or substance. It is defined as the amount of heat energy required to increase the temperature of one kilogram of the material by one degree Celsius.

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.

A hot pan has a mass of \(0.5kg\), a specific heat capacity of \(500 \frac{J}{kg^{\circ}C}\), and a current temperature of \(78^{\circ}C\). This pan comes in contact with a colder plate with a mass of \(1kg\), a specific heat capacity of \(0.323 \frac{J}{kg^{\circ}C}\), and a current temperature of \(12 ^{\circ}C\).

Using the equation above and ignoring other forms of heat loss, what will the temperature of both objects once thermal equilibrium is reached?

First thing we need to is plug our variables into the equation:

\[0.5 \cdot 500 \cdot (T_{e} - 78)+1 \cdot 0.323 \cdot (T_{e} - 12)=0\]

At this point, we can multiply all of our terms together to get this:

\[(250T_{e} - 19,500) + (0.323T_{e} - 3.876)=0\]

We then combine our terms containing T_{e} and put our other values to the other side of the equation, like so:

\[250.323T_{e}=19,503.876\]

Finally, we divide on one side to get our value of temperature at equilibrium:

\[T_{e}=77.91^{\circ}C\], to 2 decimal places.

Not much of a change for our pan, and a big change for our plate! This is due to the specific heat capacity of the plate being far lower than that of the pan, meaning its temperature can be changed much more by the same amount of energy. An equilibrium temperature that is between both of the initial values is what we are expecting here - if you get an answer that is higher than the hotter temperature, or colder than the cooler temperature, then you've done something wrong in your calculations!

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.

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!

Thermal Equilibrium - Key takeaways

• Thermal equilibrium is a state two objects interacting thermally can reach when they are at the same temperature with no net heat energy transferred between them.
• Thermal equilibrium involves temperature at a molecular level, and the transference of kinetic energy between molecules.
• An equation to solve to find the thermal equilibrium temperature is \(m_{h}c_{h}(T_{e}-T_{h_{c}})+m_{c}c_{c}(T_{e}-T_{c_{c}})=0\)
• There are many examples of thermal equilibrium in everyday life, such as thermometers and changes of state.