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Chapter 10: Problem 75

Determine whether each of the following changes will increase, decrease, or not affect the rate with which gas molecules collide with the walls of their container: (a) increasing the volume of the container, (b) increasing the temperature, (c) increasing the molar mass of the gas.

Short Answer

Expert verified
In summary, (a) increasing the volume of the container will decrease the rate of collisions between gas molecules and container walls, (b) increasing the temperature will increase the rate of collisions, and (c) increasing the molar mass of the gas will not significantly affect the rate of collisions, as the decrease in collision frequency is offset by the increase in the force of collisions.

Step by step solution

01

Effect of Volume on Collision Rate

When the volume of the container increases, the gas molecules will have more space to move around. This means they are less likely to collide with the walls of the container. Therefore, increasing the volume of the container will decrease the rate with which gas molecules collide with the container walls.
02

Effect of Temperature on Collision Rate

As the temperature of the gas increases, the average kinetic energy of the gas molecules increases as well. According to the Kinetic Molecular Theory, gas molecules move faster at higher temperatures. Faster gas molecules will collide more frequently and with greater force. Thus, increasing the temperature of the gas will increase the rate of collisions between gas molecules and the container walls.
03

Effect of Molar Mass on Collision Rate

Now let's consider increasing the molar mass of the gas. The relationship is less straightforward, but we can use the ideal gas law, PV = nRT, where P is the pressure, V is the volume, n is the number of moles, R is the gas constant, and T is the temperature, to analyze. Given the same pressure, volume, and temperature, increasing the molar mass means having more massive gas molecules in the container. When the gas molecules are more massive, they will have higher momentum and kinetic energy at the same temperature. However, at the same temperature, massive gas molecules will move more slowly than lighter gas molecules, since the kinetic energy is inversely proportional to the mass. Therefore, the more massive gas molecules will collide with the container walls less frequently but with greater force. The overall rate with which gas molecules collide with the walls will not be significantly affected, as the decrease in collision frequency is offset by the increase in the force of collisions. So, increasing the molar mass of the gas will not affect the rate of collisions with the container walls.

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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 is essential to understanding gas behavior. It proposes that gas molecules are in constant, random motion and their speed depends on their temperature. Like tiny billiard balls, these molecules are constantly colliding with each other and the walls of their container.
  • Gas molecules are small compared to the space between them.
  • No significant forces attract or repel individual molecules.
  • Collisions are elastic, meaning they don't lose energy upon impact.
This theory helps us explain behavior changes in gases when conditions like temperature and volume change. For instance, if you increase the volume of a container, the molecules have more room to spread out. This makes them less likely to hit the walls, reducing collision rates. If you increase temperature, molecules move faster, hitting walls more often and with more force, highlighting the impact of temperature on collision rates.
effect of temperature on gases
Temperature plays a crucial role in determining how gas molecules behave. When temperature rises, the kinetic energy of gas molecules increases, causing them to move faster. This is a direct consequence of the Kinetic Molecular Theory. Faster trade means more frequent collisions with the container walls.
  • Higher temperatures lead to more energetic collisions.
  • The frequency of collisions increases with rising temperature.
  • The force of each collision is more significant due to increased molecular speed.
This means that increasing the temperature of a gas typically raises the rate at which the gas molecules collide with the container walls. Conversely, if you lower the temperature, molecules slow down, leading to reduced collision frequency and force. The relationship between temperature and kinetic energy is linear, making comprehension intuitive, thanks to this simple theory.
ideal gas law
The ideal gas law is a crucial tool used to understand the behavior of gases, particularly in relation to their pressure, volume, and temperature. It is expressed as \(PV = nRT\), where:
  • \(P\) represents the pressure of the gas
  • \(V\) is the volume of the gas container
  • \(n\) stands for the number of moles of gas
  • \(R\) is the universal gas constant
  • \(T\) is the temperature in Kelvin
This equation shows the interdependence of these properties. When changes occur in one variable, it affects the others to maintain balance. For instance, increasing the molar mass of the gas at constant temperature and pressure means you have heavier molecules which collide with the walls less frequently but with greater force.

However, since frequency diminishes and force increases, they balance each other out, causing no significant change in overall collision rate. Understanding this equation helps predict how changing conditions affect a gas' behavior, including factors such as collision rates.

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Most popular questions from this chapter

Indicate which of the following statements regarding the kinetic-molecular theory of gases are correct. (a) The average kinetic energy of a collection of gas molecules at a given temperature is proportional to \(m^{1 / 2}\) . (b) The gas molecules are assumed to exert no forces on each other. (c) All the molecules of a gas at a given temperature have the same kinetic energy. (d) The volume of the gas molecules is negligible in comparison to the total volume in which the gas is contained. (e) All gas molecules move with the same speed if they are at the same temperature.

Calculate the pressure that \(\mathrm{CCl}_{4}\) will exert at \(80^{\circ} \mathrm{C}\) if 1.00 mol occupies \(33.3 \mathrm{L},\) assuming that (a) \(\mathrm{CCl}_{4}\) obeys the ideal-gas equation; (b) \(\mathrm{CCl}_{4}\) obeys the van der Waals equation. (Values for the van der Waals constants are given in Table \(10.3 .\) ) (c) Which would you expect to deviate more from ideal behavior under these conditions, \(\mathrm{Cl}_{2}\) or \(\mathrm{CCl}_{4}\) ? Explain.

Suppose that a woman weighing 130 \(\mathrm{lb}\) and wearing high- heeled shoes momentarily places all her weight on the heel of one foot. If the area of the heel is \(0.50 \mathrm{in.}^{2},\) calculate the pressure exerted on the underlying surface in (a) pounds per square inch, (b) kilopascals, and (c) atmospheres.

The metabolic oxidation of glucose, \(\mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6},\) in our bodies produces \(\mathrm{CO}_{2},\) which is expelled from our lungs as a gas: $$\mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}(a q)+6 \mathrm{O}_{2}(g) \longrightarrow 6 \mathrm{CO}_{2}(g)+6 \mathrm{H}_{2} \mathrm{O}(l)$$ (a) Calculate the volume of dry \(\mathrm{CO}_{2}\) produced at body temperature \(\left(37^{\circ} \mathrm{C}\right)\) and 0.970 atm when 24.5 \(\mathrm{g}\) of glucose is consumed in this reaction. (b) Calculate the volume of oxygen you would need, at 1.00 \(\mathrm{atm}\) and \(298 \mathrm{K},\) to completely oxidize 50.0 \(\mathrm{g}\) of glucose.

The temperature of a \(5.00-\mathrm{L}\) 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 per second.
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