Problem 145 - General Chemistry: Principles and Modern Applications

Question

Under which conditions is \(\mathrm{Cl}_{2}\) most likely to behave like an ideal gas? Explain. (a) \(100^{\circ} \mathrm{C}\) and \(10.0 \mathrm{atm}\) (b) \(0^{\circ} \mathrm{C}\) and 0.50 atm; \((\mathrm{c}) 200^{\circ} \mathrm{C}\) and \(0.50 \mathrm{atm}\) (d) \(400^{\circ} \mathrm{C}\) and \(10.0 \mathrm{atm}\).

Step-by-Step Solution

Answer

Among the given conditions, Cl2 is most likely to behave like an ideal gas in condition (c) 200°C and 0.50 atm as this condition is at a high temperature and a low pressure.

Step 1: Identify the behavior of ideal gases

As per the kinetic molecular theory, ideal gases behave best at high temperatures and low pressures. This is because at high temperatures, the kinetic energy of the gas molecules is high, meaning they move quickly and interact less with each other. At low pressures, there is more space between the molecules, reducing the chances of intermolecular collisions.

Step 2: Apply the conditions on given scenarios

(a) In this scenario, the temperature is high, which is good for ideal gas behavior. However, the pressure is also high, which is not ideal. Therefore, Cl2 will not behave ideally under these conditions. (b) In this condition, the temperature is not high, but the pressure is low, which is ideal. However, the low temperature may cause Cl2 to not behave ideally. (c) Both the temperature is high and the pressure is low in this scenario, which are ideal conditions for an ideal gas. Thus, Cl2 is most likely to behave like an ideal gas under this condition. (d) Similar to scenario (a), the high temperature is ideal but the high pressure is not, hence Cl2 will likely not behave ideally under these conditions.

Step 3: Choose the most ideal condition

From the above explanations, it is clear that a gas behaves ideally at high temperatures and low pressures. Therefore, among the given options, condition (c) 200°C and 0.50 atm is the condition under which Cl2 is most likely to behave like an ideal gas.

Key Concepts

Kinetic Molecular Theory

The kinetic molecular theory is a foundational concept in understanding gas behavior. It describes gases as composed of many small particles that are in constant, random motion. These particles are far apart from each other relative to their size, so they rarely interact.

According to this theory, several key assumptions are made:

• Gas particles move in straight lines until they collide with something, like a container wall or another gas particle.
• These collisions are perfectly elastic, meaning that no energy is lost in the process.
• The average kinetic energy of gas particles is directly proportional to the temperature in Kelvin.

This theory helps explain why gases tend to expand to fill their containers, and how changes in temperature or pressure can affect their behavior. The kinetic molecular theory is essential for understanding both ideal and real gases, providing a basis for why gases deviate from ideality under certain conditions.

Temperature and Pressure Effects on Gases

Temperature and pressure significantly impact how gases behave, according to the ideal gas laws. At higher temperatures, gas molecules gain kinetic energy, moving faster and spreading apart. This can diminish intermolecular forces, leading gases to behave more ideally.

Lowering pressure also contributes to ideal gas behavior. With fewer molecules in a given space, there are less frequent collisions between them. The result is less interaction and a closer approximation to ideal conditions.

• High temperature = higher kinetic energy = less molecular interaction.
• Low pressure = fewer collisions = less influence from intermolecular forces.

Understanding these effects explains why condition (c) in the exercise (200°C and 0.50 atm) results in behavior closest to that of an ideal gas. It represents an environment where the kinetic molecular theory can most effectively apply.

Real Gases vs Ideal Gases

In the real world, gases often deviate from ideal behavior. Real gases have intermolecular forces and occupy space, contrary to the perfect conditions described by the kinetic molecular theory. At high pressures and low temperatures, these deviations become more pronounced.

There are several reasons for these deviations:

• Intermolecular forces become more significant in real gases, affecting their movement and collision.
• Gas particles occupy a finite volume, unlike the assumptions for ideal gases.

In contrast, at high temperatures and low pressures, as assumed in ideal conditions, these factors have less effect. Hence, real gases can behave more like ideal gases under these circumstances. By understanding these differences, one can better predict how gases will act under various conditions, such as in the given exercise.