The Ideal Gas Law: Which Substances Apply?

The ideal gas law applies to pure gases and, under certain conditions, to real gases. The ideal gas law is derived from a model and relies on the assumptions that gas particles have negligible volume, are equally sized, move randomly, and have perfect elastic collisions with no energy loss or gain. No gas has these properties, but the behaviour of real gases can be described quite closely by the ideal gas law.

The ideal gas law applies when gases are subjected to either very low pressures or high temperatures. This is because low-pressure systems allow gas particles to experience fewer intermolecular forces, and high-temperature systems allow gas particles to move quickly, exhibiting fewer intermolecular forces.

The ideal gas law also applies to a system containing multiple ideal gases, known as an ideal gas mixture. However, it does not apply when gases are near their condensation point, and it does not account for chemical reactions in the gaseous phase.

Liquids

The ideal gas law applies to pure gases or pure liquids. However, it does not apply to liquids.

The ideal gas law can be used with saturated gases and saturated liquids, provided they are not two-phase substances. The ideal gas law is derived from a model and, like any other model, it applies where its underlying assumptions are good approximations of reality.

For a gas to be considered ideal, the following four assumptions must be true:

• The gas particles have negligible volume compared to the total volume of gas.
• The gas particles are equally sized and do not have intermolecular forces, such as attraction or repulsion, with other gas particles.
• The gas particles move randomly in agreement with Newton's laws of motion, which describe kinetic energy.
• The gas particles have perfect elastic collisions with no energy loss or gain.

Although no gas has these properties, the behaviour of real gases is described quite closely by the ideal gas law. Real gases behave ideally when subjected to either very low pressures or high temperatures. This is because low-pressure systems allow gas particles to experience fewer intermolecular forces, and high-temperature systems allow gas particles to move quickly, resulting in fewer intermolecular forces.

Therefore, some liquids can exhibit the properties of ideal gases.

High-pressure systems

The ideal gas law is a simplified equation of state for a hypothetical ideal gas, which can be used to describe the behaviour of many gases under certain conditions. The law is expressed as:

> PV = nRT

Where P is pressure, V is volume, n is the amount of substance (or number of moles), R is the ideal gas constant, and T is absolute temperature.

The ideal gas law is based on several assumptions, including that gas particles have negligible volume, are equally sized, and do not have intermolecular forces. However, these assumptions do not hold true for real gases, which are subject to intermolecular forces and occupy a finite volume. Therefore, the ideal gas law is most accurate for monatomic gases at high temperatures and low pressures, where the potential energy due to intermolecular forces is less significant compared to the particles' kinetic energy.

At high pressures, the ideal gas model tends to fail as the volume of a real gas is often considerably larger than that of an ideal gas. This is because, at higher pressures, the assumption that the average distance between molecules is much larger than the size of the molecules no longer holds. The deviation from ideal behaviour can be described by the compressibility factor, Z.

Despite its limitations, the ideal gas law is still widely used and applicable to a range of substances, including many gases such as nitrogen, oxygen, hydrogen, noble gases, and some heavier gases like carbon dioxide. It is also applicable to certain mixtures, such as air.

Low-temperature systems

The ideal gas law applies to a wide range of gases under many conditions, although it has some limitations. Real gases can be considered ideal in low-pressure or high-temperature systems.

In low-temperature systems, the ideal gas model tends to fail as intermolecular forces and molecular size become more significant. This is due to the low volume occupied by the gas, which increases the impact of these factors. However, the ideal gas law can still be useful in certain low-temperature scenarios. For example, it can be used to calibrate anesthetic mixtures, which are commonly used in anesthetics to ensure patient safety. The law can also be applied to model the behaviour of certain plasmas and gaseous mixtures, such as dusty plasma particles, as well as to study surface tension in water.

In reality, ideal gases do not exist, and the ideal gas law is a theoretical concept. However, it is a valuable tool for understanding and predicting gas behaviour in various contexts, including clinical and engineering applications.

Mixtures of gases

The ideal gas law applies to mixtures of gases. This is because the ideal gas law is independent of the type of gas. The ideal gas law states that PV = nRT, where P is the absolute pressure of a gas, V is the volume it occupies, n is the number of moles of the gas, R is the universal gas constant, and T is its absolute temperature.

The ideal gas law can be derived from basic principles but was originally deduced from experimental measurements of Charles's law and Boyle's law. In the ideal gas model, the volume occupied by its atoms and molecules is a negligible fraction of V. The ideal gas law describes the behaviour of real gases under most conditions.

The ideal gas law can be used to calculate the number of moles of gas in a given volume. For example, let's consider a bike tire with a volume of 2.00 x 10^-3 m^3, a pressure of 7.00 x 10^5 Pa, and a temperature of 18.0°C. We can rearrange the ideal gas law to solve for n and substitute the known values to find that there are 0.579 moles of gas in the tire.

The ideal gas law can also be used to calculate pressure changes due to temperature changes. For example, suppose a bicycle tire is fully inflated with an absolute pressure of 7.00 x 10^5 Pa at a temperature of 18.0°C. We can use the ideal gas law to calculate the pressure after the temperature has risen to 35.0°C, assuming no leaks or changes in volume. By rearranging the equation and substituting the known values, we find that the final pressure is approximately 7.41 x 10^5 Pa, which is about 6% greater than the original pressure.

The ideal gas law is also useful for calculating the number of molecules in a given volume of gas. For example, let's calculate the number of molecules in a cubic meter of gas at standard temperature and pressure (STP). Using the ideal gas law, we find that there are approximately 2.68 x 10^25 molecules in one cubic meter of gas at STP. This number is huge, even in small volumes. For instance, 1 cm^3 of gas at STP contains 2.68 x 10^19 molecules.

The ideal gas law can be restated using moles instead of the number of molecules. This is done by multiplying and dividing the equation by Avogadro's number, NA. Avogadro's number is the number of molecules in one mole of a substance, and its value is 6.02 x 10^23 mol^-1. The ideal gas law in terms of moles is PV = nRT, where n is the number of moles and R is the universal gas constant. The numerical value of R in SI units is approximately 8.31 J/mol·K.

Gases with intermolecular forces

The ideal gas law applies to a hypothetical ideal gas, which assumes that there are no intermolecular attractions between the molecules or atoms of the gas. However, in reality, gases can exhibit intermolecular forces, which are electrostatic in nature and include van der Waals forces and hydrogen bonds.

Van der Waals forces are a general term for the attraction of intermolecular forces between molecules. They are further categorised into two types: weak London dispersion forces and stronger dipole-dipole forces.

London dispersion forces, also known as instantaneous dipole-induced dipole interactions, occur in nonpolar molecules or atoms. They are caused by temporary fluctuations in the electron distributions within these entities, resulting in the formation of short-lived instantaneous dipole moments. These instantaneous dipoles can interact with the electrons in adjacent atoms or molecules, creating induced dipoles and attractive forces. The energy of these interactions falls off rapidly with increasing distance, following a 1/r^6 relationship. Larger atoms tend to exhibit stronger London dispersion forces due to their greater polarizability, which is the ease with which their electron distribution can be perturbed.

Dipole-dipole interactions, on the other hand, occur in polar molecules with net dipole moments. These molecules tend to align so that the positive end of one dipole is near the negative end of another, creating attractive interactions. The energy of dipole-dipole interactions is proportional to 1/r^3, where r is the distance between dipoles, making them substantially weaker than ion-ion interactions.

Hydrogen bonds are a special type of dipole-dipole interaction that occurs when a hydrogen atom is bonded to a highly electronegative atom, such as oxygen, nitrogen, or fluorine. The large difference in electronegativity results in a large partial positive charge on hydrogen and a corresponding negative charge on the adjacent atom. This creates strong dipole-dipole interactions, with hydrogen acting as the donor and the electronegative atom as the acceptor. Hydrogen bonds are responsible for the unusually high boiling point of water and the unique structure of ice, which is less dense than liquid water.

In summary, while the ideal gas law assumes no intermolecular forces, real gases can exhibit various types of intermolecular forces, including London dispersion forces and dipole-dipole interactions, with hydrogen bonds being a specific example of the latter. These forces play a crucial role in determining the physical properties of gases, such as their boiling points and structure.

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