In topic B.1 Thermal energy transfers we met the concepts of the molecular model and internal energy. In this topic we will apply those concepts to idealised gases and explore the consequences. We will examine in more detail how the macroscopic properties of a gas are related to the behaviours of individual molecules. We will consider the simplifications and assumptions that allow us to develop universal laws for the behaviour of gases and finally apply those models and laws to make predictions and explain observed phenomena.
The molecular model when applied to gases is known as the kinetic model. The basic assumptions of this model are:
Gases are made from molecules. These molecules are tiny, smooth, solid particles with insignificant volume, therefore the volume of the gas itself is made up almost entirely of empty space.
All collisions are elastic as described by the laws of mechanics (in topic A.2 Forces and momentum). No energy is gained or lost in the collision.
Collisions take very little time, compared to the time between collisions for a particular molecule.
There are no intermolecular forces between the molecules except during collisions.
As a consequence of these assumptions there is a range of molecular speeds (described by the Maxwell-Boltzmann distribution) and directions.
A substance obeying these assumptions is known as an ideal gas. Due to the final assumption, ideal gases can never form a liquid or a solid when cooled or compressed. Real gases are clearly not the same as ideal gases, particularly when significantly cooled and compressed. This is due to the presence of intermolecular forces and, at particularly cold temperatures quantum effects. However, warm, low pressure and relatively rarefied (low density) gases do approximate well to an ideal gas.
According to the assumptions of the kinetic model, as described above, our gas consists of free floating particles moving in straight lines that only deviate from those straight lines when they encounter each other, or the wall of the container, at which point they bounce of in an elastic collision.
As you have learned in topic A.2, an elastic collision with a wall means a change of momentum for the particle. From Newton's second law to change momentum requires a force, so the particle will experience a force, and from Newton's third law, every force has a force pair on the other object - so there must be a force on the wall. For a given area of a wall, over a given amount of time, there will be a total force exerted. We can use this to determine the pressure of the gas on the wall using the equation:
pressure = force / area, P = F / A
where the units of force are newtons (N), the units of area are square metres (m²) and the units of pressure are therefore N/m². This SI derived unit is also known as a pascal, with the symbol Pa.
It is important when simulating gases to know how much gas we are dealing with. The unit for this measurement is the mole (symbol, mol). A mole is counting word and is equivalent to, by definition, 6.022 140 76 × 10^23 particles. This number is known as Avogadro's number, and is given the symbol NA. Thus, the amount of substance is described using the equation:
n = N / NA
where n is the amount of the gas in moles, N is the number of molecules of the gas and NA is Avogadro's number = 6.022 140 76 × 10^23.
This isn't a very useful equation as written as it is impossible to count individual atoms. However, when dealing with actual gases we can find the number of moles with reference to the periodic table. For example, suppose we have a sample of oxygen gas. We know the molecular formula for oxygen gas is O₂, and the mass of each atom is 18 u so the molecular mass is 32 u. The molecular mass in u (atomic mass units) is equivalent to one mole of the substance in grams, so a mole of oxygen gas has a mass of 32g. If we measure the mass of our oxygen sample and find it is 64g then we know we have 2 moles of oxygen molecules (and just over 12 × 10^23 molecules).
We need to be able to describe the amount of a gas in moles, because it is part of the equation of state for gases.
Gases are relatively simple. For an ideal gas all we need to know to completely describe a gas are quantities:
P: the gas pressure, described above, and measured in pascals (Pa) or N/m²
V: the volume of the gas, in m³.
n: the amount of the gas, described above, measure in moles (mol).
T: the temperature of the gas, measured in kelvin (K).
Assuming we do not change the amount of the gas, n, in the sample, changing any one of these quantities (e.g. the temperature, by heating the gas) will change one or both of the other quantities, pressure and/or volume. As a relationship we can demonstrate (more detail below) that:
P V / T = a constant
Including a constant of proportionality and the amount of the gas in moles, the full equation describing the relationship between all of the quantities that describe a gas, the equation of state for the gas is:
P V = n R T
This is known as the Ideal Gas Law. R, called the ideal gas constant, is the constant of proportionality which has a value of 8.31 J/K/mol. In other words, how much energy is needed to raise the temperature of 1 mole of an ideal gas by 1 K.
This relationship was developed empirically through a series of investigations conducted through the 17th and 18th centuries that looked at the relationship between pairs of these variables. In brief these were:
PV = constant (known as Boyle's Law, temperature is held constant)
V/T = constant (known as Charles's Law, pressure is held constant)
P/T = constant (known as Gay-Lussac's Law, volume is held constant)
An alternative formulation for the ideal gas law is:
P V = N kB T
In this version, N is the absolute number of molecules (not in moles) and kB is Boltzmann's constant = 1.38 × 10^−23 J / K. This is because we define the gas constant as the product of Avogradro's number and the Boltzmann's constant: R = NA kB
Boltzmann's constant describes how much extra energy (in joules) a gas particle gains per kelvin of temperature and, since the 2019 redefinition of the SI base units, it has an exact value of 1.380 649 × 10^−23 J / K. Avogadro's number, NA, now has an exact value of 6.022 140 76 × 10^23, meaning R has an exact value of 8.314 462 618 153 24 J / K / mol.
The changes of state represented by the ideal gas law and through the individual gas laws can be represented through pressure-volume diagrams.
Continuing our look at the pressure of a gas, we can derive a relationship between the pressure a gas an exerts and the velocity of those particles by going back to momentum theory:
Suppose our sample of gas, with N molecules, is enclosed in a cube with sides of length L. The gas has a density of ρ.
An individual gas molecule has an initial velocity, u. When it collides with a wall (call it face A of the cube) then it will experience a change in momentum, ∆ p, of 2mu, with m being the mass of the molecule.
After colliding with face A it will travel back across the cube, collide with the opposite face, bounce back and eventually collide with face A again. The time taken to do this will be t = 2L/u. t is the period between impacts.
Therefore the number of impacts of the molecule on face A per unit time (the frequency of the impacts) will be the inverse of t. 1/t = u/2L.
From Newton's 2nd law, force = rate of change of momentum, ∆p/∆t = 2mu / (2L/u) = mu²/L. To recap, the force on one face by one molecule will be mu²/L.
Pressure = force / area, and the area of face A is L². So the pressure on face A is P = F/A = (mu²/L) / L² = mu²/L³. But L³ is the volume of the cube, so this can be rewritten as P = mu²/V.
However, there are N molecules in total and, on average, only a third of them will be bouncing of face A (since this is a cube). So the pressure from all the molecules will be P = 1/3 × m/V × Nv² where v is the mean square (or average) velocity in all directions, not just the direction bouncing off face A.
Rewriting (7) we get P = 1/3 × mN/V ×v². Since mN is the total mass of the gas, M, and the density of the gas is ρ = M/V. We can rewrite again to give us the relationship:
P = 1/3 × ρ v²
This means that the pressure of a gas is directly proportional to the density of a gas and also directly proportional to the average velocity squared of the molecules. As the average velocity squared of the particles is also proportional to their kinetic energy (since EK = 1/2 × mv²) and temperature is also directly proportional to the average kinetic energy, this is another way of demonstrating that P/T is a constant (Gay-Lussac's law).
We saw as part of topic B.1 Thermal energy transfer that the relationship between the average kinetic energy of gas molecules and their temperature in kelvin is described using the equation:
EK = 3/2 × kB T
For a gas the only component of the internal energy, U, is the kinetic energy. There are no bonds so there is no potential energy. EK in the equation above is the average kinetic energy per molecule, so if there are N molecules in the gas in total then the total kinetic energy of the sample, and therefore the total internal energy in total will be given by the equation:
U = 3/2 × N kB T
As we saw when looking at the equation of state, N = n NA (the total number of molecules = the number of moles of molecules × Avogadro's number) and R = NA kB (the gas constant equals Avogadro's number × Boltzmann's constant), so this equation can also be written as:
U = 3/2 × R n T
Image credits:
Banner NASA/ESA/NOIRLab/NSF/AURA/M.H. Wong and I. de Pater (UC Berkeley) et al.Acknowledgments: M. Zamani, CC BY 4.0 <https://creativecommons.org/licenses/by/4.0>, via Wikimedia Commons