Question

At the same temperature, rank the following substances in order of increasing average velocity: \(\mathrm{CO}_{2}, \mathrm{He}, \mathrm{H}_{2}, \mathrm{CH}_{4}\).

Short answer

The gases in order of increasing average velocity are: CO2, CH4, He, H2

Step-by-step solution

Determine the molar masses

Refer to the periodic table and compile the molar masses of each gas molecule:- For \(CO_2\), Carbon has a molar mass of 12g/mol and Oxygen has a molar mass of 16g/mol. Thus, \(CO_2\) with one Carbon atom and two Oxygen atoms has a molar mass of \(12 + 2*16 = 44g/mol\).- For \(He\), Helium only consists of one atom and thus has a molar mass of 4g/mol.- For \(H_2\), Hydrogen has a molar mass of 1g/mol so \(H_2\) with two Hydrogen atoms has a molar mass of \(2*1 = 2g/mol\).- For \(CH_4\), Carbon has a molar mass of 12g/mol and Hydrogen has molar mass of 1g/mol. Hence, \(CH_4\) with one Carbon atom and four Hydrogen atoms has a molar mass of \(12 + 4*1 = 16g/mol\).

Rank the substances

Now rank the substances by their molar masses:1. \(He\) (4g/mol)2. \(H_2\) (2g/mol)3. \(CH_4\) (16g/mol)4. \(CO_2\) (44g/mol)Due to the kinetic theory of gases, the one with the least molar mass will have the highest average velocity, so invert this list for the ranking order.

Key concepts

Kinetic Theory of Gases

The kinetic theory of gases provides a foundational explanation for the behavior of gas particles based on their motion. It postulates that a gas is composed of tiny particles in constant, random motion. These gas molecules collide with each other and the walls of their container, exerting pressure. The temperature of the gas reflects the average kinetic energy of these molecules. As temperature increases, the average velocity of gas molecules also increases since kinetic energy is directly related to velocity.

The theory also explains that at a given temperature, all gases have the same average kinetic energy. This reveals why lighter gas molecules move faster than heavier ones; energy distributed to a smaller mass results in a greater velocity. Therefore, for gases at the same temperature, the average velocity is inversely proportional to the square root of its molar mass.

Molar Mass

Molar mass is a critical property of substances, defined as the mass of a given substance (chemical element or chemical compound) divided by its amount of substance. It is typically expressed in units of grams per mole (g/mol). Knowing the molar mass of a gas is crucial when comparing the behavior of different gaseous substances since it affects various physical properties under the same conditions. For instance, molar mass influences the rate of diffusion and effusion of gases as described by Graham's law, and it is inversely related to gas molecule velocity per the kinetic theory of gases.

To calculate the molar mass of a compound, like those mentioned in the exercise, you must sum up the molar masses of each constituent element, taking into account their respective quantities in a single molecule of the compound. This calculation, as shown in the problem's step-by-step solution, lays the groundwork for understanding how molar mass affects gas behavior.

Gas Molecule Velocity Ranking

Gas molecule velocity ranking is an essential concept in comparing the behavior of different gases at the same temperature. Based on the kinetic theory of gases, one can determine which gas particles will move faster by looking at their molar masses. Lighter molecules, with lower molar masses, will have higher average velocities while heavier molecules move slower.

In the context of the given exercise, once the molar masses are determined, students should be able to rank the gases in order of increasing average velocity by inversely relating these molar masses to the velocities. A common mistake in such problems is to directly relate molar mass with velocity rather than considering the inverse relationship. Thus, the correct order from fastest to slowest will be the reverse of the molar mass ranking: Helium (He), Hydrogen (H2), Methane (CH4), and then Carbon Dioxide (CO2). This ranking is instrumental in predicting and explaining the behavior of gases in different scenarios, such as when they are used in gas chromatography, or when estimating rates of molecular diffusion in air.