Question

A real gas closely approaches the behaviour of an ideal gas at (a) 15 atmospheres and \(220 \mathrm{~K}\) (b) 1 atm and \(273 \mathrm{~K}\) (c) \(0-5 \mathrm{~atm}\) and \(500 \mathrm{~K}\) (d) 15 atm and \(500 \mathrm{~K}\)

Short answer

The condition in which a real gas behaves most closely to an ideal gas is when the pressure is \(0 - 5 \mathrm{~atm}\) and the temperature is \(500 \mathrm{~K}\).

Step-by-step solution

Analyze the given conditions

The conditions given are different combinations of temperature and pressure. Our task is to identify the combination where a real gas behaves most closely to an ideal gas. According to the Kinetic Theory of Gases, real gases behave most like ideal gases at high temperatures and low pressures.

Apply the Theory

Out of the given options, let's map the theory on each combination: (a) 15 atmospheres and \(220 \mathrm{~K}\): High pressure and low temperature, so the gas is likely to behave less ideally. (b) 1 atm and \(273 \mathrm{~K}\): Low pressure and relatively higher temperature, so the gas is likely to behave more ideally. (c) \(0-5 \mathrm{~atm}\) and \(500 \mathrm{~K}\): Very low pressure and high temperature, so the real gas is most likely to behave like an ideal gas in this condition. (d) 15 atm and \(500 \mathrm{~K}\): High pressure and high temperature, so the gas is likely to behave less ideally due to high pressure.

Conclude the result

Using the principle that gases tend to behave as ideal gases under high temperature and low pressure conditions, we can conclude that option (c) \(0-5 \mathrm{~atm}\) and \(500 \mathrm{~K}\) would be the condition in which a real gas would most closely mimic the behaviour of an ideal gas.

Key concepts

Kinetic Theory of Gases

The Kinetic Theory of Gases is a fundamental concept that helps us understand the behavior of gases on a molecular level. This theory makes a few simple assumptions to explain how gases act.

• Gas molecules are in constant, random motion.
• The collisions that occur between gas molecules are elastic, meaning that there is no net loss of energy.
• The actual volume occupied by gas molecules is negligible compared to the volume of their container.
• The forces between molecules are minimal, especially in ideal gases.

These assumptions help predict that, under certain conditions, real gases can behave like ideal gases. To do so, the gas must have:
- High temperature - Low pressure These conditions ensure that the energy and spacing of the gas particles are optimized to minimize interactions and approximate the behavior described by the kinetic theory. By knowing this, we can predict and explain phenomena in various scientific and industrial applications where understanding gas behavior is crucial.

High Temperature

When gas molecules are at high temperatures, their kinetic energy is increased. This translates into molecules moving faster and colliding more vigorously. In the context of real vs ideal gases under these conditions:

• Molecules have more energy to overcome intermolecular forces.
• Increased molecular motion reduces the effect of any small attractive forces between particles.

This means that the behavior of a gas at higher temperatures tends to lean closer to that of an ideal gas.
Ideal gases do not "stick" to one another due to intermolecular forces, as these forces are assumed to be negligible.
Hence, higher temperatures help shun the reality of molecular attractions, enabling real gases to more closely resemble their ideal counterparts. This is why, in the exercise, conditions with temperatures of 500 K improve the likelihood of real gases behaving ideally.

Low Pressure

Low pressure conditions have significant implications for gas behavior in terms of ideal gas approximation. At low pressures, molecules are spread out over a larger volume.

• The number of collisions between molecules is reduced.
• The distance between molecules means fewer interactions and lesser influence from intermolecular forces.

This aligns with the assumptions of the kinetic theory, ensuring minimal intermolecular force interactions. Quieter collisions mean that molecules are less likely to enter into interaction-rich scenarios, resulting in behavior that aligns more closely with the assumptions of the ideal gas law. In the exercise, gases at pressures between 0-5 atm are better at mimicking ideal gases because of this very effect; there are fewer opportunities for interactions that would deem a gas "non-ideal."