Question

When the pressure of a sample of gas is increased from \(0.50\) to \(100 \mathrm{~atm}\) at constant temperature, its volume decreases from \(2.01\) to \(13 \mathrm{ml}\). What could cause the deviation from Boyle's law? (a) Volume of the gas molecules is a significant fraction of the volume of container at higher pressure. (b) The force of attraction between the gas molecules is greater when the pressure is higher. (c) The molecules are dimerized at the higher pressure. (d) The collision of the molecules on the walls of the container are no longer elastic at higher pressure.

Short answer

The deviation from Boyle's Law could be due to the volume of gas molecules becoming significant compared to the volume of the container (a), or the intermolecular forces becoming stronger at higher pressure (b). Dimerization (c) and non-elastic collisions (d) could also contribute to the deviation.

Step-by-step solution

Analyzing Boyle's Law

Boyle's Law states that for a given amount of gas at constant temperature, the pressure of the gas is inversely proportional to its volume. Mathematically, it can be expressed as PV = constant. This law is based on the assumption that gases consist of tiny particles in random motion and that the volume of the gas particles is negligibly small compared to the volume of their container. It also assumes no intermolecular forces and that collisions between molecules are perfectly elastic.

Identifying Causes of Deviation

Deviations from Boyle's Law occur when the assumptions behind it break down. At very high pressures, the volume of gas molecules may become a significant fraction of the total volume of the gas. Additionally, intermolecular forces may become significant, causing the pressure to be higher or lower than predicted by Boyle's Law for the measured volume. In case (c), dimerization changes the number of particles and thus affects the law's applicability. Lastly, non-elastic collisions (d) can also result in a deviation because they would change the internal energy of the gas and thus violate the constant-temperature assumption.

Evaluating Each Option

Option (a) could cause deviation as at higher pressures the volume of the gas molecules may indeed become significant. Option (b) is also a potential cause since intermolecular forces can affect the pressure-volume relationship. Option (c), the dimerization of molecules under high pressure, could potentially affect the applicability of Boyle's law by changing the number of molecules. Option (d), which suggests the collisions are no longer elastic at higher pressure, would imply the kinetic energy of the molecules is changing, which contradicts the constant temperature assumption.

Conclusion

Options (a), (b), (c), and (d) can all cause deviations from Boyle's Law at high pressures. However, the typical and most immediate causes for deviation are options (a) - the significance of the volume of the gas molecules, and (b) - the force of attraction between molecules becoming significant.

Key concepts

Pressure-Volume Relationship

Understanding the pressure-volume relationship is essential when studying the behavior of gases. At the heart of this concept is the idea that gas volume decreases as pressure increases when temperature is held constant. This inverse correlation is a cornerstone of Boyle's Law, which can be represented by the equation, \[ PV = k \], where \( P \) is pressure, \( V \) is volume, and \( k \) is a constant for a given amount of gas at a fixed temperature.

The pressure-volume relationship assumes that gases are ideal. This means the gas particles do not take up space themselves and do not interact with one another. However, real gases do occupy space and can interact under certain conditions, leading to deviations from this simple relationship. When faced with such deviations, we can explore variables like the size of gas particles and intermolecular forces to understand the non-ideal behavior of gases.

Gas Laws

Gas laws describe how the physical properties of gases change in relation to each other. Alongside Boyle's Law, which addresses the pressure-volume relationship, other principal laws include Charles's Law, relating volume and temperature, and Avogadro's Law, which links volume and the amount of gas (number of moles) at constant temperature and pressure. The Ideal Gas Law, \[ PV=nRT \], incorporates Boyle's, Charles's, and Avogadro's laws to describe the state of an ideal gas.

However, real gases often deviate from ideal behavior due to factors like intermolecular attractions and the finite volume of gas particles. Understanding each law's implications and limitations furthers our grasp of how gases will react to different conditions in scientific and real-world applications.

Intermolecular Forces

Intermolecular forces are the forces of attraction and repulsion that act between neighboring particles: molecules, atoms, and ions. These forces, including van der Waals forces and hydrogen bonds, are significant in determining the physical properties and behaviors of substances. In the context of gas laws, when pressure increases, molecules are forced closer together, and the effect of these intermolecular forces becomes more pronounced.

As a result, the assumptions of gases acting ideally — with no intermolecular forces — begin to break down. At high pressures, as in our exercise, these forces can sufficiently alter the pressure-volume relationship to cause observable deviations from Boyle's Law, especially since gases are more easily compressed and can have their volume significantly affected by these forces.

Non-Ideal Gas Behavior

While Boyle's Law provides a foundational perspective on gas behavior, it assumes ideal conditions that are not always met in real-world scenarios. Non-ideal gas behavior arises when the assumptions of negligibly small particle sizes and no intermolecular forces fail. This may occur under conditions of high pressure or low temperature, where gases are compressed to the point where the particles' own volumes are non-negligible, and their attractions or repulsions impact the gas's properties. Real gases may also display non-elastic collisions, which defies the presumption that internal energy remains constant with changing pressure.

It's key to recognize the factors that cause a gas to behave non-ideally to accurately predict and understand these behaviors. Corrections to the Ideal Gas Law, like those from the Van der Waals equation, attempt to account for this non-ideal behavior by considering particle volume and intermolecular forces.