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Chapter 10: Problem 83

The temperature of a 5.00-L container of \(\mathrm{N}_{2}\) gas is increased from \(20^{\circ} \mathrm{C}\) to \(250^{\circ} \mathrm{C}\). If the volume is held constant, predict qualitatively how this change affects the following: (a) the average kinetic energy of the molecules; (b) the root-mean-square speed of the molecules; (c) the strength of the impact of an average molecule with the container walls; (d) the total number of collisions of molecules with walls ner second.

Short Answer

Expert verified
In summary, when the temperature of a \(5.00-L\) container of \(\mathrm{N}_{2}\) gas is increased from \(20^{\circ} \mathrm{C}\) to \(250^{\circ} \mathrm{C}\), while holding the volume constant, the following changes will occur: (a) the average kinetic energy of the molecules will increase; (b) the root-mean-square speed of the molecules will increase; (c) the strength of impact of an average molecule with the container walls will increase; (d) the total number of collisions of molecules with walls per second will increase.

Step by step solution

01

Understanding the Concepts

To begin with, according to the kinetic theory of gases, the average kinetic energy of gas molecules is directly proportional to the temperature of the gas (measured in Kelvin). Also, the root-mean-square speed of gas molecules is dependent on the temperature and molecular mass. The strength of impact of molecules with the container walls and the total number of collisions with the walls per second are linked to the pressure exerted by the gas, which is determined by the number of molecules, temperature and volume of the container.
02

(a) Predicting the change in Average kinetic energy of molecules

As temperature increases, the average kinetic energy of the gas molecules also increases. This is because the gas molecules absorb heat energy, which gets transformed into kinetic energy causing them to move faster. So, the average kinetic energy of the molecules will increase.
03

(b) Predicting the change in root-mean-square speed of the molecules

The formula for the root-mean-square speed of gas molecules is given by \(\sqrt{3kT/m}\), where \(k\) is Boltzmann’s constant, \(T\) is the temperature in Kelvin and \(m\) is the molecular mass. As temperature increases, it would cause the root-mean-square speed of the molecules to increase. So, the root-mean-square speed of the molecules will increase.
04

(c) Predicting the change in strength of impact of an average molecule with the container walls

The strength of impact of molecules on the container walls is related to the pressure the gas exerts on the walls. As temperature increases, the molecules have more kinetic energy and move faster, striking the container walls with more energy. Because the volume is constant, this increase in molecule speed and energy results in an increase in pressure, thus increasing the strength of impact. So, the strength of impact of an average molecule with the container walls will increase.
05

(d) Predicting the change in the total number of collisions of molecules with walls per second

Again, with the increase in temperature, the speed of molecules increases. Faster molecules means they will hit the container walls more frequently, because they take less time to travel back and forth within the container. So, the total number of collisions of molecules with walls per second will increase.

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