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Chapter 3: Problem 52

At constant volume for a fixed number of a moles of gas, the pressure of the gas increases with the rise in temperature due to (a) increase in average molecular speed (b) increase in rate of collisions (c) increase in molecular attraction (d) increase in mean free path

Short Answer

Expert verified
(a) Increase in average molecular speed and (b) increase in rate of collisions lead to higher pressure.

Step by step solution

01

Read the Question Carefully

The exercise asks us to determine why the pressure of a gas increases with a rise in temperature at constant volume and a fixed number of moles. We have four options to consider: increase in average molecular speed, rate of collisions, molecular attraction, and mean free path.
02

Understand the Relationship Between Pressure and Temperature

When a gas is heated, its molecules move faster. According to the kinetic theory of gases, the pressure of a gas is related to the kinetic energy of the molecules, which depends on their speed. Therefore, as temperature increases, the average speed of the molecules increases, impacting pressure.
03

Analyze Each Option

(a) Increase in average molecular speed: This directly affects pressure as increased speed means more forceful impacts with the container walls. (b) Increase in rate of collisions: As temperature rises, molecules move faster, leading to more collisions, which contributes to increased pressure. (c) Increase in molecular attraction: Not significant for ideal gases as these are considered negligible in the ideal gas law, and would typically decrease pressure due to cohesion. (d) Increase in mean free path: A higher temperature doesn't increase mean free path; it would remain constant at a fixed volume.
04

Select the Correct Answer

The main factors leading to increased pressure when temperature increases at constant volume are options (a) an increase in average molecular speed and (b) an increase in rate of collisions. These are closely related and both contribute to the increase in pressure.

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Key Concepts

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

Average Molecular Speed
In the realm of kinetic theory, the average molecular speed is a measure of how quickly molecules move within a gas. This speed tends to increase when the temperature rises and the volume stays constant.
When the temperature of a gas increases, it means that the thermal energy supplied to the gas also rises. This energy is translated into kinetic energy, causing gas molecules to move faster.
Consequently, with faster speeds, molecules will collide more forcefully with the walls of the container. These frequent and vigorous collisions are pivotal for understanding changes in pressure.
Pressure-Temperature Relationship
The pressure-temperature relationship in gases is fundamentally linked through the kinetic energy of particles. When the temperature of a gas is increased, its particles gain kinetic energy and move with greater speed.
This enhanced movement results in an increased rate of collision of gas molecules against the container walls, leading to higher pressure.
The kinetic theory of gases explains this concept:
  • The pressure of a gas is caused by collisions of particles with the walls of its container.
  • As the temperature increases, the frequency and force of these collisions increase, thereby increasing pressure.
This relationship is crucial in understanding why gases behave the way they do under various temperatures. The more we heat a gas at constant volume, the more it wants to "expand" by increasing pressure.
Ideal Gas Law
The ideal gas law is an equation of state that describes how pressure, volume, and temperature interact for an ideal gas. It is succinctly expressed as: \[ PV = nRT \] where \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the universal gas constant, and \( T \) is temperature.
This equation can help us see how pressure is directly proportional to temperature when volume and moles of gas are constant.
  • A rise in temperature means more kinetic energy for the particles.
  • In a fixed volume, this increase in kinetic energy results in higher pressure.
The ideal gas law simplifies the complex dynamics of gas behavior into a more manageable form, helping us predict how changes in temperature affect gas pressure.

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Most popular questions from this chapter

The densities of two gases are in the ratio of \(1: 16\). The ratio of their rates of diffusion is (a) \(16: 1\) (b) \(4: 1\) (c) \(1: 4\) (d) \(1: 16\)

Equal weights of methane and oxygen are mixed in an empty container at \(25^{\circ} \mathrm{C}\). the fraction of the total pressure exerted by oxygen is (a) \(1 / 2\) (b) \(2 / 3\) (c) \(1 / 3 \times 273 / 298\) (d) \(1 / 3\)

A 2 mole mixture of \(\mathrm{Ne}(\mathrm{g}), \mathrm{H}_{2}(\mathrm{~g})\) and \(\mathrm{O}_{2}(\mathrm{~g})\) are placed in a closed container at a pressure equal to \(50 \mathrm{~atm}\). An electric spark is passed and pressure noted is \(12.5 \mathrm{~atm}\) after cooling. Oxygen gas is introduced for pressure to become 25 atm. Again electrical spark is passed and pressure drops to \(10 \mathrm{~atm}\). (all measurements are at same T and P) (a) Moles fraction of \(\mathrm{O}_{2}\) in original mixture is \(0.25\) (b) Moles of \(\mathrm{O}_{2}\) added after first spark is \(0.5\) (c) Mole fraction of \(\mathrm{H}_{2}\) in original mixture is \(0.7\) (d) Mole fraction of Ne in original mixture is \(0.005\)

A bottle of dry ammonia and a bottle of dry hydrogen chloride connected through a long tube are opened simultaneously at both ends, the white ammonium chloride ring first formed will be (a) at the centre of the tube (b) near the hydrogen chloride bottle (c) near the ammonia bottle (d) throughout the length of the tube

The most probable speeds of the molecules of gas \(\mathrm{A}\) at \(\mathrm{T}_{1} \mathrm{~K}\) and gas \(\mathrm{B}\) at \(\mathrm{T}_{2} \mathrm{~K}\) are in the ratio \(0.715: 1\). The same ratio for gas \(\mathrm{A}\) at \(\mathrm{T}_{2} \mathrm{~K}\) and gas \(\mathrm{B} \mathrm{T}_{1} \mathrm{~K}\) is \(0.954\). Find the ratio of molar masses \(\mathrm{M}_{\mathrm{A}}: \mathrm{M}_{\mathrm{B}}\). (a) \(1.965\) (b) \(1.0666\) (c) \(1.987\) (d) \(1.466\)
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