The particle nature of matter is the model for matter that is based on the idea that all matter is composed of microscopic particles. Current science tells us that these particles are molecules, atoms, ions, etc, but the idea goes back to ancient Greek models of reality and most of the content of this topic is based on 18th and 19th century ideas that provided the bases of the modern industrial age and were the spark for more modern models such as nuclear and quantum physics.
In B.1 Thermal energy transfers we will look at how the microscopic models of matter are based on macroscopic observations. We will look at the importance of energy transfers between part of a system and how they lead to changes in the system, and we will see how the observations we can make of the properties of a system can let us predict other properties of a system.
Molecular theory describes how the large-scale properties of solids, liquids and gases can be explained and predicted using the idea that they are all made from microscopic particles we call molecules. You will know from previous science that molecules are built from atoms, but any changes to molecules are the result of chemical changes and this specific section is only concerned with physical changes - changes of state - rather than chemical reactions.
In a solid we need to consider the intermolecular bonds (the electrostatic forces) between the molecules (not the chemical bonds between the atoms that make up the molecules). In a solid these bonds act like springs connecting the molecules. These bonds hold the molecules of the solid in place relative to each other, allowing the molecules to vibrate in place, but keeping the overall shape of the solid.
In a liquid the bonds are weaker. These reduced forces are strong enough to keep the molecules close to each other, but weak enough to let the molecules pass over each other and let the overall liquid adapt the shape of the container it is in.
In a gas there are effectively no bonds between the molecules. The molecules are free to move part from each, constrained only by the barriers of a container, collisions with neighbouring particles and, on VERY large scales, the force of gravity. Because of the differing nature of the bonds between the molecules, in both liquids and gases (collectively known as fluids) the sample of matter takes the shape of the container, but only gases can be compressed.
Density is a measurement of the mass of a substance in a particular volume. The symbol for density is the Greek letter rho (ρ) with the equation being density = mass / volume, in symbols ρ = M / V. Density is an SI derived scalar quantity with standard units of kilogrammes per metre cubed, kg / m³.
As well as depending on the type of material, the density of a substance also depends on its temperature and the pressure exerted upon it. For most substances heating increases the volume and therefore decreases the density. Increasing the pressure tends to decrease the volume and therefore increase the density. We will look at these relationships for the case of gases in more detail later in the unit.
Most solids have a greater density than liquids. There are some exceptions, mercury is a liquid at room temperature and pressure and has a very high density, while some solids such as pumice, cork or aerogels have very low densities. Gases always have less density than solids and liquids.
The process of floating is dependent on density. If a solid, liquid or gas has an average density less than the density of the medium it is in, then it will float on or in that medium. A ship will float on water as long as its average density (including the air spaces inside the ship) is less than that of the water (i.e. less than 1000 kg/m³). A balloon will rise in the air as long as its average density is less than the surrounding air (about 1.225 kg/m³ at sea level). The density of air decreases with height, so the balloon will rise only until its has the same density as the air outside.
We all understand temperature as how 'hot' or 'cold' a substance is. The first attempts to quantify and invent temperature scales came with the development of the first thermometers at the end of the 17th century. The most common temperature scale in everyday use is the Celsius scale. Like all temperature scales, it is defined by two fixed points and a number of graduations between them. For Celsius this was traditionally defined as the freezing and boiling point of water at standard pressure with 100 degrees separating the two.
First defined in 1954, the Kelvin scale has increments with same magnitude as the Celsius scale. An increase or decrease of 1 kelvin (1 K) is the same magnitude as an increase or decrease of 1 degree celsius (1 ºC). However, the fixed points of the Kelvin scale are absolute zero, the coldest possible temperature and the triple point of water (effectively the freezing point) at standard pressure.
Since the magnitude of 1 K is the same as 1 ºC, converting between the two is a simple formula:
T (in kelvin) = T (in degrees Celsius) + 273
T (in degrees Celsius) = T (in kelvin) - 273
Although both Celsius and kelvin are commonly used, the official SI unit of thermodynamic temperature is the kelvin. Since 2019, the kelvin is defined in terms of fundamental units, primarily the Boltzmann constant (of which more later).
In all states of matter, the particles are able to move relative to each other, either through vibration or through absolute motion. This random motion is separate to any overall motion of the object (for example, the motion of particles in a tennis ball is independent from the motion of the tennis ball itself). Thus, due to this random motion, the particles have kinetic energy, measured in joules (J). Although every particle in an object will be moving at a different velocity, there will generally be a distribution of velocities, and therefore energies, and we can calculate an average velocity or energy. Note that in a substance with different particles of different masses, the average velocities of the different types of particles will be different, but they will have the same distribution of energies.
The temperature of a substance is proportional to the average random kinetic energy of the particles in the substance.
Mathematically this relationship is described by the Boltzmann equation: the average kinetic energy (EK) of each particle in joules = 1.5 × the Boltzmann constant × the temperature of the substance in kelvin. The Boltzmann constant (kB) equals 1.38 × 10^-23 J / K and represents the amount of energy needed to raise the temperature of a single particle by 1 kelvin. In symbols:
EK = 3/2 × kBT
The temperature of an object will determine the direction of heat transfer between two objects. Heat energy (see below) will always be transferred from a hotter object (making it cooler) to the colder object (making it hotter) until the two objects reach an equilibrium temperature, the same temperature for both of them. The transfer itself will take place by one or more methods of heat transfer, namely conduction, convection or radiation. Again we will look at these concepts in detail later in this unit.
As we have seen, a substance includes particles, separated (except in a gas) by bonds. Each molecule has an average amount of kinetic energy, determined by the temperature of the substance, and if we multiply this average kinetic energy with the number of molecules then we will have a total kinetic energy of all the particles in the substance. The intermolecular bonds between the molecules posses a certain magnitude of potential energy, energy due to position. The sum of all of the kinetic energy of the molecules in a substance (due to the movement of the molecules) and the potential energy (due to the bonds between those molecules) is called the total internal energy of the object or substance. Internal energy is given the symbol U and is measured in joules, J.
The internal energy of a substance can change because it is doing work (for example, a gas expanding and pushing a piston) or through heat transfer. The energy transferred between objects is known as heat or heat energy. Note that heat is always a description of energy transfer between substances, rather than the internal energy or average kinetic energy of a particular object.
Any transfer of energy (heat transfer) between objects will increase or decrease the internal energy of the object depending on the direction of the heat transfer, which is always from higher temperature to lower temperature. The total amount of internal energy lost by one object(s) must equal the internal energy gained by the other object(s), due to conservation of energy. However, changes to internal energy can change the average random kinetic energy OR the internal potential energy of a substance. Normally the average kinetic energy of the particles will change, i.e. the temperature will change, but at certain temperatures the potential energy will change, due to a change of state, also known as a phase change.
If a substance receives or gives away heat energy and it is not at its melting or boiling point, then the temperature of the substance will change. The relationship between the magnitude of the energy transferred (in joules) and the change in temperature (in kelvin) is determined by the mass of the substance (in kilograms) and a property of the substance itself called the specific heat capacity of the substance.
The specific heat capacity (shc, symbol c) of a substance is the amount of energy, in joules, needed to raise the temperature of 1 kilogram of a substance by 1 kelvin. Alternatively it is the heat energy (in joules) released by a substance, per kilogram of the substance, when it cools by one kelvin.
Heat energy transferred = mass of substance (in kg) × specific heat capacity (in J / kg / K) × change in temperature (in K or ºC)
Heat energy transferred is given the symbol Q, so in symbols this equation is:
Q = mc∆T
For example, the specific heat capacity of liquid water is approximately 4200 J / kg / K. For every kilogram of water it needs 4200 J to increase the temperature by 1 K (or 1 ºC). So to increase the temperature of 2 litres of water by 20 kelvin would need 168 kilojoules of energy. Two litres of water has a mass of 2 kg, so heat energy transferred = 2 kg × 4200 J/kg/K × 20 K = 168 000 J.
The phase of a substance is a particular uniform arrangement of the molecules in a substance. Every state of matter (solid, liquid and gas) is a different phase. In addition to the states of matter there are additional phases (for example, different arrangements of water molecules in different types of solid ice) but for this topic we will generally deal just with the solid, liquid and gaseous phases and the transitions between them. For consistency we will call these transitions melting (solid to liquid, heat given to the liquid), freezing (liquid to solid, heat taken from the liquid), vaporisation (liquid to gas, heat given to the gas) and condensation (gas to liquid, heat taken from the gas). The transitions from solid straight to gas and vice-versa (known as sublimation and deposition) are not covered in this topic.
As mentioned above, a phase change will occur if heat energy is transferred between objects and one or both the objects is at a particular temperature. For melting and freezing this temperature is called the melting point. For vaporisation and condensation this temperature is the boiling point. (Evaporation is a separate issue and is not dealt with in this topic). The melting point and boiling point of a substance depend on what the substance is and other conditions such as the pressure on the substance and/or the presence of contaminants.
During the process of changing phase the temperature of an object does not change. In the cases of melting and vaporisation, the heat energy from the hotter substance is being used to increase the separation of the particles, acting against the bonds, and increasing the internal potential energy of the cooler substance. Thus the internal energy increases, because the potential energy is increasing, but the average kinetic energy of the particles is not increasing and thus the temperature does not change until the process is complete.
Similarly, during condensation or freezing, the potential energy of the substance decreases (and thus so does the internal energy) but the average kinetic energy of the substance does not change, and again the temperature remains the same until the process is complete.
The amount of energy (in joules) required to change the phase of a unit mass (one kilogram) of a substance is known as latent heat. For melting (or freezing) this is known as the specific latent heat of fusion. The vaporisation (or condensing) this is known as the specific latent heat of vaporisation. The specific heat of vaporisation is always greater than the specific heat of fusion because it is more difficult to separate the molecules from the liquid to the vapour phase than from the solid to the liquid phase.
Specific latent heat has the symbol L and units of joules per kilogram (J / kg). In symbols, the equation for this relationship is:
Q = mL
For water the specific heat of fusion is 334 000 J/kg and the specific heat of vaporisation is 2 260 000 J/kg. Thus if we have 20 kg of solid ice at 0 ºC we will need (20 × 334 000 =) 6 680 000 J to turn it into 20 kg of liquid water at 0 ºC. The same 20 kg of water will need (20 x 2 260 000 =) 45 200 000 J to be turned from 20 kg of liquid water at 100 ºC to 20 kg of water vapour at 100 ºC, assuming standard pressure.
Heat always flows from hot substances to cold substances, and as we said previously, there are three primary methods of heat transfer. These are conduction, convection and radiation.
Conduction occurs in solids and at the boundary between solids and other states of matter. Conduction does not occur within fluids or a vacuum. In a solid the molecules are able to move only by vibrating about a fixed point. The higher the temperature of a solid, the faster those molecules vibrate. If one part of a solid is heated, then the molecules will be vibrating faster than the other molecules in the solid. However, as the molecules are connected by their spring-like bonds then these vibrations will be passed along. As the molecules further from the heat source start to vibrate more, then their average kinetic energy will increase, in other words those parts of the substance will increase in temperature. This is heat transfer by conduction.
The rate at which heat transfer by conduction takes place (given the symbols ∆Q/∆t, in units of joules per second) between two different parts of a conductor depends on the cross-sectional area of the substance in the direction of conduction (symbol A, units m²), the length of the conductor (symbol x, unit of metres), the difference in temperature (symbol ∆ T, units of kelvin or ºC) and a property of the solid called the conductivity of the solid (given the symbol k, with units of W/m/K or W/m/ºC). An increase in the cross-sectional area or the temperature difference will increase the rate of heat conduction, while an increase in length will decrease the rate of heat conduction. The full expression is:
∆Q/∆t = k.A.∆T/∆x
All solids conduct, but some solids conduct better than others. The conductivity of metals is particularly high due to the presence of free electrons in metallic bonding. These free electrons are able to pass on their kinetic energy much more quickly (though collisions with the lattice ions) than the simple method of linked vibrations through the bonds described above.
Convection is a method of heat transfer that occurs in fluids (i.e. liquids and gases). Traditional convection does not occur in a vacuum or in solids, although over very long periods plastic materials such as hot mantle rocks deep within the Earth can also undergo convection-like processes. Convection occurs because hotter fluids expand and therefore become less dense. As we saw in the section on density, a less dense material will rise when surrounded by a more dense medium. Thus the hot fluid rises, taking its heat energy with it. As it does it mixes with the medium, exchanging heat with it, until it reaches equilibrium and there is no longer a temperature (or density) difference. Measnwhile, the rising hot fluid is replaced by surrounding cold fluid. If the heat source is still present then this cold fluid will become hot and the process repeats, possibly leading to convection currents.
Heat transfer by radiation occurs because all objects above absolute zero emit electromagnetic (EM) waves. Note that the nature and some properties of EM waves will be discussed in more detail in the waves topic of this course. For this section it is enough to know that EM radiation can described as waves that propagate in electric and magnetic fields. They are described in terms of their wavelength (given the symbol λ, the lower-case Greek letter lambda, and with units of metres (m)). The full range of possible wavelengths is called the Electromagnetic Spectrum and it is divided into regions. The regions that concern us most in this topic are the infrared (wavelengths from about 1 mm down to 700 nm) and the visible (from about 700 nm down to 450 nm). Since EM waves are a form of energy, then emitting EM radiation reduces the internal energy of a substance and absorbing EM radiation increases the internal energy of a substance, making this a form of heat transfer. Since EM radiation can pass through any transparent medium it can transfer through a vacuum.
Radiation occurs across all wavelengths, but (for a perfect emitter, known as a black-body) the peak wavelength will depend on the temperature. This wavelength distribution is known as the emission spectrum of the hot object. For cold and about room temperature objects all of the emitted radioan will be in the part of the spectrum known as the infra-red. For objects about a few hundred degrees celsius this radiation will start to include significant parts of the visible spectrum, and the object will start to visibly glow, For an object as hot as the Sun (with a surface temperature of almost 6000 K) about half of the emitted radiation is in the visible and the peak wavelength is in the middle of the visible section of the spectrum making the Sun seem white to our eyes. The relationship between the temperature of the hot radiating object and the peak wavelength of the radiation is described by Wien's displacement law:
λ(max).T = 2.9×10^-3 mK
where λ(max) is the peak wavelength (in metres) and T is the surface temperature of the object in kelvin.
The amount of EM radiation emitted by an object per second is known as the luminosity of the object. The symbol for luminosity is L and the units are joules/second or watts (W). EM radiation is emitted from the surface of an object, so the greater the surface area (symbol A, units m²) of the object then the greater the luminosity. However the greatest factor in determining the luminosity of an object is its temperature (symbol T, unit kelvin (K)). The equation describing this relationship is known as the Stefan-Boltzmann law and in full it is:
L = σAT⁴
where the Greek letter sigma (σ) represents the Stefan-Boltzmann constant, 5.67×10^-8 W/m²/K⁴.
Note that the Stefan-Boltzmann law describes the rate of heat transfer emitted by a perfect emitter, also known as a black-body. Most materials are not perfect emitters, especially not across the full range of wavelengths emitted. Stars are one example of a reasonably good black-body emitter. The luminosity of stars is often described in terms of the luminosity of the Sun. The luminosity of our Sun (L☉) is about 3.828×10^26 W.
All objects emit their heat radiation in all directions. Therefore the radiation absorbed by an object will depend not only on the luminosity of the source (symbol L, unit watts (W)), but also on the distance separating the source and the absorbing object. This distance we will give the symbol d and the unit will be metres (m). We will normally assume a equal distribution of that emission in all directions, thus the it is the inverse distance squared (1/d²) that determines the relationship, an application of the inverse-square law.
The amount of radiation emitted by the emitting object that is absorbed by the absorbing object is known as the apparent brightness of the emitting object (apologies if that sounds confusing, read it again). Apparent brightness is given the symbol b and has unit of watts per metre squared. The bigger the absorbing object than the more radiation it will absorb. The equation expressing this relationship is:
b = L / 4.𝛑.d²
If we know the luminosity of the source and we measure its brightness then we can determine the distance. This application is used a lot to determine the distance to astronomical objects.
Image credits:
Banner Heat Roger Smith Flickr https://www.flickr.com/photos/rogersmith/88419650