Thermal Physics

We’re all naturally familiar with the concept of temperature, but how do we actually define a substance’s temperature using physics? Describing the internal thermal energy of a substance using a temperature turns out to be a fundamental aspect of physics and thermodynamics.

What is temperature?

The thermal energy of a substance is proportional to the (average) kinetic energy of its constituent molecules or atoms. In a system with two objects, the hotter object has a higher temperature and higher average kinetic energy. Thermal energy flows from the hotter object to the colder one until both objects reach the same temperature. This is formalised in the zeroth law of thermodynamics.

The zeroth law of thermodynamics

Although the zeroth law of thermodynamics was only the fourth law to be proposed, it was deemed so fundamental to thermal physics that it now comes first.

The zeroth law states:

This (seemingly obvious) law defines temperature as a property that describes the direction of any thermal energy transfer between objects. The zeroth law is important as it shows that the transfer of thermal energy is controlled by physical temperatures rather than an object’s total thermal (kinetic) energy.

Temperature scales

Having defined what temperature is, we now need a way to measure it. In order to compare the temperatures of different objects, a scale is needed. A temperature scale is defined by two fixed points at specific temperatures, with a number of increments between them.

There are currently three main temperature scales in use around the world:

Celsius

• This temperature scale, proposed by Swedish astronomer Anders Celsius in 1742, is used by most of the world.
• The fixed points used by the Celsius scale are the freezing point (0°C) and the boiling point of water (100°C at atmospheric pressure 1.01⋅10⁵ Pa), with 100 increments of 1°C between them.

Fahrenheit

• The Fahrenheit scale, proposed by German physicist Daniel Fahrenheit in the 18th century, is primarily used in the United States.
• The scale is also based on the freezing (32°F) and boiling (212°F) points of water, with 180 increments between them.

• The absolute temperature scale uses the fixed points of the triple point of water (273.16°K) and absolute zero (0°K). These were selected because they do not vary with atmospheric pressure, unlike the Celsius and Fahrenheit scales.
• When defining the Kelvin scale, it was decided that each increment should be equal to 1°C to make comparisons simpler. This is why there are exactly 273.16°F between the two fixed points.
• The kelvin is the SI unit of temperature. To convert between temperatures in Celsius and kelvin, we can use the formula T(K)=T(C)+273.16.
• Temperatures on the kelvin scale are always positive.

Heat transfer

Now that we have an understanding of what temperature is, we can explore how thermal energy (heat) is transferred between objects in a system. This field of physics is known as thermodynamics, which deals with the relationships between heat, work, temperature, and energy in systems.

The laws of thermodynamics

We have already encountered the zeroth law of thermodynamics, which provides the basis for the definition of temperature. Let us now explore the remaining laws of thermodynamics.

The first law of thermodynamics

Q is used to represent the change in thermal energy in the first law of thermodynamics, which states:

\[\Delta U = Q -W\]

This shows that energy cannot be created or destroyed.

The second law of thermodynamics

The second law introduces the property of entropy, represented by S.

A unit of heat transferred (δQ) is a product of the temperature of the systems (T) and the change in total entropy (∂s).

\[\delta Q = T \cdot ∂ S\]

The third law of thermodynamics

The third law states that, as a system’s temperature approaches absolute zero and all thermal energy is removed, the system reaches a constant ground state. The value of entropy at this point is known as the system residual entropy. If the system only has one possible microstate at absolute zero, then the residual entropy will also be zero. The constant value of a system’s residual entropy at absolute zero increases with the number of possible system microstates.

\[S = k_B \cdot \ln(\Omega)\]

Here, S is the system residual entropy, k_B is the Boltzmann constant, and Ω is the number of microstates.

A pure crystal is an example of a material that would have zero residual entropy, as there is only one valid crystalline structure (microstate) its atoms can adopt.

Internal energy

All substances (solid, liquid, or gas) have internal energy (U), which is the sum of the kinetic energy and potential energy of its constituent molecules. The levels of kinetic and potential energy each individual molecule has are random (within the range of energy contained in the system), but we can understand these properties of the larger substance by using the average energies of its molecules.

The kinetic energy of molecules in a substance is directly related to its temperature, while their potential energy depends on the phase of the substance. The potential energy is made up of electrostatic potential energy. The attractive forces between molecules are ‘stretched’ as the average distance between molecules increases when transitioning from solid to liquid, or liquid to gas. This increases their electrostatic potential energy, just as increasing the distance between an object and the earth raises its gravitational potential energy.

The diagram shows how energy is transferred into a substance at a steady rate. During phase changes, the kinetic energy of its molecules (temperature) remains constant, while the electrostatic potential energy increases. This is because, during the phase change, the input energy is transferred to an increase in electrostatic potential as the substance melts or evaporates and forces between the molecules change.

Outside of phase changes, the electrostatic potential remains relatively constant, and all the input energy is converted into molecular kinetic energy, increasing the temperature of the substance. The total internal energy is a sum of these two components and increases at the same rate as energy is transferred into the substance.

• In solids, the electrostatic potential energy is a large negative, as the strong bonds between atoms or molecules require significant energy to be broken.
• In liquids, the electrostatic forces between molecules are smaller, so the potential is a smaller negative, as less energy is required to break them.
• In gases, the electrostatic potential is near zero, as the forces between molecules are very weak or negligible.

Latent heat

The contribution of electrostatic potential energy to the total internal energy creates a property known as the specific latent heat of a substance.

Specific heat

Different substances require varying amounts of thermal energy transfer to change their temperatures by the same amount. Water is an example of a substance with a high specific heat capacity (think of how long it takes to boil water for a cup of tea). This is because a large amount of energy needs to be transferred into the water by the kettle to raise its temperature to 100°C. An example of a low specific heat capacity is iron, and therefore most steels (think of how quickly a steel spoon heats up in the mug of tea you just made).

The specific heat capacity of a substance is calculated as:

Change in energy = mass · specific heat capacity · change in temperature

\[\Delta E = mc \Delta \theta\]

Here, ΔE is the change in internal energy, m is the mass of substance, c is the specific heat capacity, and Δθ is the change in heat energy. Specific heat capacity is given in units of J⋅kg^-1K^-1.

Ideal gases

We can describe macroscopic properties, such as the mass, temperature, or pressure of gases, relatively easily. However, in order to fully understand how gases behave, we also need to know what’s happening at the level of an individual particle. The number of atoms (or molecules) in a volume of gas is described using a unit called mole, the SI unit for the amount of substance, which indicates the number of atoms or molecules in a given sample of a given substance.

The Avogadro constant can be used to calculate the number of atoms N in n moles of a substance:

\[N = n \cdot N_A\]