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Chapter 10: Problem 78

Indicate which of the following statements regarding the kineticmolecular theory of gases are correct. For those that are false, formulate a correct version of the statement. (a) The average kinetic energy of a collection of gas molecules at a given temperature is proportional to \(\mathrm{m}^{1 / 2}\). (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.

Short Answer

Expert verified
(a) False. Correct statement: "The average kinetic energy of a collection of gas molecules at a given temperature is proportional to the temperature." (b) True. (c) False. Correct statement: "The gas molecules at a given temperature exhibit a distribution of kinetic energies that follow the Maxwell-Boltzmann distribution." (d) True. (e) False. Correct statement: "The gas molecules at a given temperature exhibit a distribution of speeds that follow the Maxwell-Boltzmann distribution."

Step by step solution

01

Statement (a)

The statement says that the average kinetic energy of a collection of gas molecules at a given temperature is proportional to \(m^{1/2}\), where m is the mass of gas molecules. This statement is false. According to the kinetic molecular theory of gases, the average kinetic energy of a collection of gas molecules at a given temperature is proportional to the temperature: \[\overline{KE} \propto T\]A correct version of the statement would be: "The average kinetic energy of a collection of gas molecules at a given temperature is proportional to the temperature."
02

Statement (b)

The statement says that gas molecules are assumed to exert no forces on each other. This statement is true. One of the assumptions of the kinetic molecular theory of gases is that there are no intermolecular forces acting between gas molecules. This simplifies the understanding of the fascinating behavior of gases.
03

Statement (c)

According to the statement, all molecules of a gas at a given temperature have the same kinetic energy. This statement is false. The kinetic molecular theory of gases assumes that the distribution of kinetic energy of gas molecules at a given temperature follows the Maxwell-Boltzmann distribution. This means that gas molecules at a given temperature can have a variety of kinetic energies, not necessarily the same. A correct version of the statement would be: "The gas molecules at a given temperature exhibit a distribution of kinetic energies that follow the Maxwell-Boltzmann distribution."
04

Statement (d)

The statement says that the volume of the gas molecules is negligible in comparison to the total volume in which the gas is contained. This statement is true. In the kinetic molecular theory of gases, it is assumed that the volume occupied by gas molecules is insignificant compared to the total volume in which the gas is contained. This allows the gas molecules to be considered as point particles.
05

Statement (e)

The statement says that all gas molecules move with the same speed if they are at the same temperature. This statement is false. As mentioned earlier, gas molecules at a given temperature have a range of kinetic energies, which implies that they have a variety of speeds as well. As a consequence, the gas molecules do not have the same speed at a given temperature. A correct version of the statement would be: "The gas molecules at a given temperature exhibit a distribution of speeds that follow the Maxwell-Boltzmann distribution."

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Most popular questions from this chapter

A piece of dry ice (solid carbon dioxide) with a mass of \(5.50 \mathrm{~g}\) is placed in a 10.0 - \(\mathrm{L}\) vessel that already contains air at 705 torr and \(24^{\circ} \mathrm{C}\). After the carbon dioxide has totally vaporized, what is the partial pressure of carbon dioxide and the total pressure in the container at \(24{ }^{\circ} \mathrm{C} ?\)

A 4.00 -g sample of a mixture of \(\mathrm{CaO}\) and \(\mathrm{BaO}\) is placed in a 1.00-L vessel containing \(\mathrm{CO}_{2}\) gas at a pressure of 730 torr and a temperature of \(25^{\circ} \mathrm{C}\). The \(\mathrm{CO}_{2}\) reacts with the \(\mathrm{CaO}\) and \(\mathrm{BaO},\) forming \(\mathrm{CaCO}_{3}\) and \(\mathrm{BaCO}_{3} .\) When the reaction is complete, the pressure of the remaining \(\mathrm{CO}_{2}\) is 150 torr. (a) Calculate the number of moles of \(\mathrm{CO}_{2}\) that have reacted. (b) Calculate the mass percentage of \(\mathrm{CaO}\) in the mixture.

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 atm and \(298 \mathrm{~K}\), to completely oxidize \(50.0 \mathrm{~g}\) of glucose.

(a) What conditions are represented by the abbreviation STP? (b) What is the molar volume of an ideal gas at STP? (c) Room temperature is often assumed to be \(25^{\circ} \mathrm{C}\). Calculate the molar volume of an ideal gas at \(25^{\circ} \mathrm{C}\) and 1 atm pressure.

(a) If the pressure exerted by ozone, \(\mathrm{O}_{3}\), in the stratosphere is \(3.0 \times 10^{-3}\) atm and the temperature is \(250 \mathrm{~K}\), how many ozone molecules are in a liter? (b) Carbon dioxide makes up approximately \(0.04 \%\) of Earth's atmosphere. If you collect a 2.0 - \(\mathrm{L}\) sample from the atmosphere at sea level \((1.00\) atm \()\) on a warm day \(\left(27^{\circ} \mathrm{C}\right),\) how many \(\mathrm{CO}_{2}\) molecules are in your sample?
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