The Ideal Gas Law
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You should be comfortable with the definitions of pressure, volume, and absolute temperature (Kelvin), and you should recall the individual empirical gas laws (Boyle, Charles, Gay-Lussac) from the previous lecture.
Why this matters
The ideal gas law unifies three separate empirical observations into a single, predictive equation.
While Boyle, Charles, and Avogadro each described how one variable changes when others are held constant, their laws are fragments of a larger truth. The ideal gas law, \(PV = nRT\), synthesises these proportionalities into a comprehensive state equation. This single relationship allows chemists to predict the behaviour of gases under any combination of conditions, provided the gas behaves ideally. It serves as the foundational bridge between the macroscopic properties we can measure (pressure, volume, temperature) and the microscopic reality of molecular motion, forming the basis for all subsequent thermodynamic and kinetic molecular theories.
The Unified Picture
Imagine you have a sealed container filled with gas. In the previous lecture, we looked at this system in isolation: if you squeeze it, the pressure rises (Boyle); if you heat it, the volume expands (Charles); if you add more gas, the volume increases (Avogadro). These were separate rules for separate situations. The ideal gas law tells us that these rules are not independent; they are different faces of the same underlying relationship. The pressure, volume, amount of substance, and temperature are all linked in a single equation. If you know any three of these variables, you can calculate the fourth. This is powerful because it means you do not need to memorise separate rules for every possible scenario. Instead, you use one universal equation. The constant \(R\) is the "universal" gas constant, which acts as the proportionality factor that makes the units work out. It is the same for all ideal gases, reflecting the fact that, at the molecular level, all ideal gases behave identically regardless of their chemical identity, provided they are at the same temperature and pressure. This universality is a profound insight: it suggests that the behaviour of a gas is determined by the number of particles and their kinetic energy, not by the specific type of particle.
Combining the Proportionalities
To understand how the individual laws merge, we start with the joint proportionality. Boyle’s law states that volume is inversely proportional to pressure (\(V \propto 1/P\)) when temperature and amount are fixed. Charles’s law states that volume is directly proportional to absolute temperature (\(V \propto T\)) when pressure and amount are fixed. Avogadro’s law states that volume is directly proportional to the number of moles (\(V \propto n\)) when temperature and pressure are fixed. By combining these three relationships, we find that volume is proportional to the product of moles and temperature, divided by pressure: \(V \propto \frac{nT}{P}\). To turn this proportionality into an equality, we introduce a constant of proportionality, \(R\). This gives us \(V = R \frac{nT}{P}\), which rearranges to the familiar form \(PV = nRT\). The key here is that \(R\) is not arbitrary; it is a physical constant derived from the behaviour of an ideal gas. Its value depends only on the units chosen for pressure, volume, and temperature. This unification shows that the gas laws are not disjointed rules but a coherent system where changing one variable necessitates a predictable change in the others, governed by this single constant.
The Universal Gas Constant
The gas constant \(R\) is a bridge between the macroscopic world of pressure and volume and the microscopic world of energy. Its value is approximately \(8.314\ \mathrm{J\,mol^{-1}\,K^{-1}}\) when using SI units (Pascals for pressure, cubic metres for volume). However, in chemistry, we often use atmospheres for pressure and litres for volume. In this case, the value of \(R\) changes to \(0.08206\ \mathrm{L\cdot atm\,mol^{-1}\,K^{-1}}\). It is crucial to match the units of \(R\) with the units of your data. If you use Pascals, you must use the SI value of \(R\). If you use atmospheres and litres, you must use the corresponding value. A common source of error is mixing units, such as using the SI value of \(R\) with pressure in atmospheres. This leads to significant calculation errors. The constant \(R\) is universal in the sense that it applies to all ideal gases, but its numerical value is unit-dependent. Understanding this allows you to solve for any unknown variable in the ideal gas equation, provided you are consistent with your units. The choice of \(R\) is dictated by the convenience of the units provided in the problem statement.
Assumptions of the Ideal Model
The ideal gas law is based on a simplified model of gas behaviour. It assumes that gas particles are point masses with no volume of their own. It also assumes that there are no intermolecular forces between the particles; they do not attract or repel each other. Finally, it assumes that collisions between particles and with the container walls are perfectly elastic, meaning no kinetic energy is lost. In reality, all gases deviate from this ideal behaviour, especially at high pressures and low temperatures. At high pressures, the volume of the particles becomes significant compared to the total volume, and the attractive forces between particles become important. At low temperatures, the kinetic energy of the particles is low enough that intermolecular attractions can cause condensation. However, for many practical purposes, particularly at low pressures and high temperatures, the ideal gas law provides an excellent approximation. Recognising the limits of this model is essential for advanced studies, but for introductory chemistry, it is the standard tool for gas calculations. The deviations from ideality are quantified by the compressibility factor, which is treated later in this unit.
The result
The ideal gas law
Three separately discovered empirical proportionalities — Boyle's law (\(V\propto1/P\) at fixed \(n,T\)), Charles's law (\(V\propto T\) at fixed \(n,P\)), and Avogadro's law (\(V\propto n\) at fixed \(T,P\), gas-stoichiometry-avogadro) — combine into a single joint proportionality \(V\propto nT/P\), which becomes an equation once a proportionality constant \(R\) (the universal gas constant, empirically found to be the same for every gas in the ideal limit) is introduced: \(PV=nRT\).
A common misconception is that the ideal gas law applies to liquids and solids as well. It does not. The law is specific to gases because the assumptions of negligible particle volume and no intermolecular forces are only valid for the gaseous state. Applying \(PV=nRT\) to a liquid will yield incorrect results because the volume of the liquid is determined by the size of the molecules and their packing, not by their kinetic energy and pressure in the same way. C
Worked example
Problem. A sample of nitrogen gas occupies a volume of \(5.00\ \mathrm{L}\) at a pressure of \(1.20\ \mathrm{atm}\) and a temperature of \(298\ \mathrm{K}\). Calculate the number of moles of nitrogen gas present.
- Identify the known variables: \(V = 5.00\ \mathrm{L}\), \(P = 1.20\ \mathrm{atm}\), \(T = 298\ \mathrm{K}\).
- Identify the unknown variable: \(n\) (number of moles).
- Select the appropriate value for the gas constant \(R\) based on the units of pressure and volume. Since pressure is in atm and volume is in L, use \(R = 0.08206\ \mathrm{L\cdot atm\,mol^{-1}\,K^{-1}}\).
- Rearrange the ideal gas law \(PV = nRT\) to solve for \(n\): \(n = \frac{PV}{RT}\).
- Substitute the values into the equation: \(n = \frac{(1.20\ \mathrm{atm})(5.00\ \mathrm{L})}{(0.08206\ \mathrm{L\cdot atm\,mol^{-1}\,K^{-1}})(298\ \mathrm{K})}\).
- Perform the calculation: \(n = \frac{6.00}{24.454} \approx 0.245\ \mathrm{mol}\).
Units check. The units of \(\mathrm{L\cdot atm}\) in the numerator cancel with the \(\mathrm{L\cdot atm\,mol^{-1}\,K^{-1}}\) in the denominator, leaving \(\mathrm{mol}\), which is the correct unit for amount of substance.
Three quick tests
- If the temperature of a gas is doubled while the pressure and volume remain constant, what happens to the number of moles?
Answer
It must halve: \(n = PV/RT\), so at fixed \(P\) and \(V\), \(n\) is inversely proportional to \(T\) (in practice, half the gas has to escape). - Why must temperature be in Kelvin when using the ideal gas law?
Answer
Because the ideal gas law is derived from absolute temperature scales; using Celsius would lead to incorrect proportionalities and negative values for volume or pressure. - If you use pressure in Pascals and volume in cubic metres, which value of \(R\) should you use?
Answer
You should use \(R = 8.314\ \mathrm{J\,mol^{-1}\,K^{-1}}\), as Joules are equivalent to \(\mathrm{Pa\cdot m^3}\).
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Every lecture and every result in this unit, with prerequisites and threads.
Lecture notes checked line by line by the editor; every number in the worked example was recomputed.