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Chapter 5: Problem 34

Consider the following samples of gases at the same temperature. Arrange each of these samples in order from lowest to highest: a. pressure b. average kinetic energy c. density d. root mean square velocity Note: Some samples of gases may have equal values for these attributes. Assume the larger containers have a volume twice the volume of the smaller containers, and assume the mass of an argon atom is twice the mass of a neon atom.

Short Answer

Expert verified
To arrange the gas samples by the attributes given: a. Pressure: Compare by the number of moles and the container volume; more moles and smaller volume lead to higher pressure. b. Average Kinetic Energy: Since the temperature is the same for all samples, their average kinetic energy will be equal. c. Density: Rank based on the product of moles and molar mass divided by the container volume; higher product and smaller volume lead to increased density. d. Root Mean Square Velocity: Compare using \( v_{rms} = \sqrt{\frac{3kT}{m}} \), considering the mass ratio between argon and neon atoms (m_argon = 2 * m_neon).

Step by step solution

01

Define variables and constants

Let's assign symbols to the variables representing pressure (P), average kinetic energy (KE), density (ρ), and root mean square velocity (v_rms). Let's also assign symbols representing the number of moles (n), volume (V), Boltzmann's constant (k), and temperature (T) for each gas sample. Assume mass (m) of the argon atom is twice the mass of a neon atom.
02

Use the ideal gas law to compare gas pressures

The ideal gas law is given by the equation: PV = nRT We want to compare the pressures of the gas samples. We assume that the temperature and the gas constant R are the same for all samples giving us: P = nR * (T/V) We know that the larger containers have a volume twice the volume of the smaller containers. Therefore, in comparing the pressures, we find that the gas with the more moles and smaller volume will have the highest pressure and vice versa.
03

Compare the average kinetic energies of gas samples

The average kinetic energy of a gas molecule is given by: KE = (3/2) * kT Since the temperature is the same for all gas samples, their average kinetic energy will also be the same.
04

Compare the densities of gas samples

Density (ρ) can be defined as mass (m) per unit volume (V). The mass of a gas sample can be calculated using the equation: m = n * M where M is the molar mass and n is the number of moles for each of the gases. Using the given information, we know that the mass of an argon atom is twice the mass of a neon atom. Therefore, the ranking of densities would depend on the product of moles and molar mass for each gas sample divided by the volume of their container.
05

Compare the root mean square velocities of gas samples

The root mean square (rms) velocity of a gas molecule is given by the equation: v_rms = \(\sqrt{(\frac{3kT}{m})}\) The temperature is the same for all gas samples, and we know the mass ratio between argon and neon atoms (m_argon = 2 * m_neon). We can now compare the samples based on their root mean square velocities. In summary, these are the key steps to ordering gas samples by various attributes. Remember to consider molar mass, molecular mass, and container volume when comparing the samples, as these factors will influence the ranking of these attributes.

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Most popular questions from this chapter

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