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Chapter 5: Problem 115

Consider a 1.0-L container of neon gas at STP. Will the average kinetic energy, average velocity, and frequency of collisions of gas molecules with the walls of the container increase, decrease, or remain the same under each of the following conditions? a. The temperature is increased to \(100^{\circ} \mathrm{C}\) . b. The temperature is decreased to \(-50^{\circ} \mathrm{C}\) c. The volume is decreased to 0.5 L . d. The number of moles of neon is doubled.

Short Answer

Expert verified
a) The average kinetic energy, average velocity, and frequency of collisions will increase when the temperature is increased to \(100^{\circ} \mathrm{C}\). b) The average kinetic energy, average velocity, and frequency of collisions will decrease when the temperature is decreased to \(-50^{\circ} \mathrm{C}\). c) The average kinetic energy and average velocity remain the same, while the frequency of collisions increases when the volume is decreased to 0.5 L. d) The average kinetic energy and average velocity remain the same, while the frequency of collisions increases when the number of moles of neon is doubled.

Step by step solution

01

a. The temperature is increased to 100掳C

When the temperature increases, the kinetic energy and average velocity of the gas molecules increase because the temperature is directly proportional to the average kinetic energy of the gas molecules. Also, as the gas molecules move faster, the frequency of collisions with the walls of the container will increase.
02

b. The temperature is decreased to -50掳C

When the temperature decreases, the kinetic energy and average velocity of the gas molecules decrease because the temperature is directly proportional to the average kinetic energy of the gas molecules. As the gas molecules move slower, the frequency of collisions with the walls of the container will decrease.
03

c. The volume is decreased to 0.5 L

When the volume decreases, the average velocity and kinetic energy of the gas molecules remain the same as the temperature is unchanged. However, since the gas molecules are confined to a smaller space, the frequency of collisions with the walls of the container will increase.
04

d. The number of moles of neon is doubled

When the number of moles of neon is doubled, the average kinetic energy and average velocity of the gas molecules will remain the same because these properties depend only on the temperature and not the number of moles. However, the frequency of collisions with the walls of the container will increase due to the increased number of gas molecules in the container.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Kinetic Molecular Theory
The Kinetic Molecular Theory provides us with a valuable understanding of the behavior of gas particles. This theory assumes that gas is composed of a large number of small particles that are in constant, random motion. Due to their motion, gas particles continuously collide with each other and with the walls of any container they are in. These collisions are considered perfectly elastic, meaning there is no loss of energy during the collisions.

There are several important assumptions involved in this theory:
  • Gas particles have negligible volume compared to the overall volume of the gas they occupy.
  • No attractive or repulsive forces exist between particles, except during collisions.
  • The average kinetic energy of gas particles is directly proportional to the absolute temperature of the gas.
Understanding these assumptions helps explain how gas changes under different conditions, such as temperature and pressure.
Temperature Effects on Gases
Changing the temperature of a gas has significant effects on its behavior. According to the Kinetic Molecular Theory, as the temperature of a gas increases, so does the average kinetic energy of its particles. This means that at higher temperatures, gas molecules move faster.

For example, if we raise the temperature of a gas from room temperature to 100掳C:
  • The average kinetic energy of the gas molecules increases.
  • The average velocity of the gas molecules increases as they move faster.
  • There will be more frequent collisions with the container walls due to the increased speed.
Conversely, if the temperature decreases, the opposite occurs: particles slow down, kinetic energy decreases, and the collision frequency diminishes. This principle underpins many gas-related processes, from cooking to industrial applications.
Volume and Pressure Relationship
The relationship between the volume and pressure of a gas is described by Boyle's Law, which states that at constant temperature, the pressure of a gas is inversely proportional to its volume. Mathematically, it is expressed as: \[ P_1V_1 = P_2V_2 \]where \(P\) denotes pressure and \(V\) denotes volume.

This means when the volume of a confined gas decreases, the pressure increases, assuming the temperature remains constant. With a decrease in volume, gas molecules are crowded into a smaller space:
  • There is less room for molecules to move, leading to more frequent collisions with the container walls.
  • The increased collision frequency results in higher pressure.
Understanding this relationship is crucial for applications like designing pressurized systems and understanding natural phenomena like weather changes.
Ideal Gas Law
The Ideal Gas Law is a foundational equation in understanding gas behavior under various conditions. It combines several gas laws, forming a single, comprehensive model. The Ideal Gas Law is expressed as:\[ PV = nRT \]where:
  • \( P \) is the pressure of the gas.
  • \( V \) is the volume of the gas.
  • \( n \) is the number of moles of the gas.
  • \( R \) is the universal gas constant.
  • \( T \) is the temperature in Kelvin.
Understanding this law helps predict how changing one variable influences others, provided the gas behaves ideally. An ideal gas perfectly adheres to the assumptions of the Kinetic Molecular Theory. While real gases might deviate under high pressure or low temperature, the Ideal Gas Law remains a powerful approximation for understanding gas behavior in most conditions.

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Most popular questions from this chapter

You have an equimolar mixture of the gases \(\mathrm{SO}_{2}\) and \(\mathrm{O}_{2},\) along with some \(\mathrm{He}\), in a container fitted with a piston. The density of this mixture at STP is 1.924 \(\mathrm{g} / \mathrm{L}\) . Assume ideal behavior and constant temperature and pressure. a. What is the mole fraction of He in the original mixture? b. The \(\mathrm{SO}_{2}\) and \(\mathrm{O}_{2}\) react to completion to form \(\mathrm{SO}_{3} .\) What is the density of the gas mixture after the reaction is complete?

Consider two different containers, each filled with 2 moles of \(\mathrm{Ne}(g)\) . One of the containers is rigid and has constant volume. The other container is flexible (like a balloon) and is capable of changing its volume to keep the external pressure and internal pressure equal to each other. If you raise the temperature in both containers, what happens to the pressure and density of the gas inside each container? Assume a constant external pressure.

An important process for the production of acrylonitrile \(\left(\mathrm{C}_{3} \mathrm{H}_{3} \mathrm{N}\right)\) is given by the following equation: $$2 \mathrm{C}_{3} \mathrm{H}_{6}(g)+2 \mathrm{NH}_{3}(g)+3 \mathrm{O}_{2}(g) \longrightarrow 2 \mathrm{C}_{3} \mathrm{H}_{3} \mathrm{N}(g)+6 \mathrm{H}_{2} \mathrm{O}(g)$$ A 150 -L reactor is charged to the following partial pressures at \(25^{\circ} \mathrm{C} :\) $$\begin{aligned} P_{\mathrm{C}, \mathrm{H}_{6}} &=0.500 \mathrm{MPa} \\\ P_{\mathrm{NH}_{3}} &=0.800 \mathrm{MPa} \\ P_{\mathrm{O}_{2}} &=1.500 \mathrm{MPa} \end{aligned}$$ What mass of acrylonitrile can be produced from this mixture \(\left(\mathrm{MPa}=10^{6} \mathrm{Pa}\right) ?\)

Helium is collected over water at \(25^{\circ} \mathrm{C}\) and 1.00 atm total pressure. What total volume of gas must be collected to obtain 0.586 g helium? (At \(25^{\circ} \mathrm{C}\) the vapor pressure of water is 23.8 torr.)

Sulfur trioxide, \(\mathrm{SO}_{3},\) is produced in enormous quantities each year for use in the synthesis of sulfuric acid. $$\mathrm{S}(s)+\mathrm{O}_{2}(g) \longrightarrow \mathrm{SO}_{2}(g)$$ $$2 \mathrm{SO}_{2}(g)+\mathrm{O}_{2}(g) \longrightarrow 2 \mathrm{SO}_{3}(g)$$ What volume of \(\mathrm{O}_{2}(g)\) at \(350 .^{\circ} \mathrm{C}\) and a pressure of 5.25 atm is needed to completely convert 5.00 \(\mathrm{g}\) sulfur to sulfur trioxide?
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