Problem 81 - Chemistry: The Central Science in SI Units, Global Edition

Question

(a) Place the following gases in order of increasing average molecular speed at \(25^{\circ} \mathrm{C}: \mathrm{O}_{2}, \mathrm{Ar}, \mathrm{CO}, \mathrm{HCl}, \mathrm{CH}_{4} \cdot(\mathbf{b})\) Calculate the rms speed of CO molecules at \(25^{\circ} \mathrm{C} .(\mathbf{c})\) Calculate the most probable speed of an argon atom in the stratosphere, where the temperature is \(0^{\circ} \mathrm{C}\).

Step-by-Step Solution

Answer

(a) The order of increasing average molecular speed at 25°C for the given gases is: Ar < HCl < CO < O₂ < CH₄. (b) The rms speed of CO molecules at 25°C is approximately 517 m/s. (c) The most probable speed of Argon atoms in the stratosphere at 0°C is approximately 490 m/s.

Step 1: (a) Ordering Gases based on Average Molecular Speeds

According to the Graham's law of effusion, the average molecular speed of a gas is inversely proportional to the square root of its molar mass. We can write this as: \(v_{1}/v_{2}=\sqrt{M_{2}/M_{1}}\) Given the gases: O₂, Ar, CO, HCl, CH₄ Let's find the molar masses for each gas: O₂: 32 g/mol Ar: 40 g/mol CO: 28 g/mol HCl: 36.5 g/mol CH₄: 16 g/mol Now order the gases based on their molar masses: CH₄ < O₂ < CO < HCl < Ar Since the average molecular speed is inversely proportional to the square root of the molar mass, the order of increasing average molecular speed is the reverse of the order of molar masses. Thus, the order of increasing average molecular speed is: Ar < HCl < CO < O₂ < CH₄

Step 2: (b) Finding the RMS Speed of CO Molecules

The formula for rms speed of molecules in a gas is: \(v_{rms} = \sqrt{3RT/M}\) where: v_rms = rms speed R = gas constant = 8.314 J/(mol K) T = temperature in Kelvin (25°C = 298.15 K) M = molar mass of the gas (CO) = 28 g/mol = 0.028 kg/mol Now we can plug the values into the equation and solve for v_rms. \(v_{rms}(CO) = \sqrt{3(8.314\;\text{J}/\text{mol K})(298.15\;\text{K}) / (0.028\;\text{kg}/\text{mol})}\) \(v_{rms}(CO) = \sqrt{3(8.314)(298.15) / (0.028)}\) \(v_{rms}(CO) ≈ 517\;\text{m/s}\) The rms speed of CO molecules at 25°C is approximately 517 m/s.

Step 3: (c) Calculating the Most Probable Speed of Argon Atoms

The formula for the most probable speed of molecules in a gas is: \(v_{p} = \sqrt{2RT/M}\) where: v_p = most probable speed R = gas constant = 8.314 J/(mol K) T = temperature in Kelvin (0°C = 273.15 K) M = molar mass of the gas (Ar) = 40 g/mol = 0.040 kg/mol Now we can plug the values into the equation and solve for v_p. \(v_{p}(Ar) = \sqrt{2(8.314\;\text{J}/\text{mol K})(273.15\;\text{K}) / (0.040\;\text{kg}/\text{mol})}\) \(v_{p}(Ar) = \sqrt{2(8.314)(273.15) / (0.040)}\) \(v_{p}(Ar) ≈ 490\;\text{m/s}\) The most probable speed of Argon atoms in the stratosphere at 0°C is approximately 490 m/s.

Key Concepts

Kinetic Molecular Theory

The Kinetic Molecular Theory is a fundamental concept that helps us understand the behavior of gases. It suggests that gas particles are in constant, random motion. This motion entails that gas molecules are always colliding with each other and with the walls of their container.
These collisions are perfectly elastic, meaning there is no loss of energy. The theory also explains that the average kinetic energy of gas particles is proportional to the temperature of the gas.
As a result, at higher temperatures, gas molecules move more rapidly, increasing both pressure and volume, given constant conditions. This relationship is essential for studying molecular speed, as it ties directly to concepts like the Root Mean Square Speed (RMS Speed) and Most Probable Speed.

RMS Speed

The Root Mean Square Speed (RMS Speed) is a measure used to describe the speed of molecules in a gas in more mathematical terms. RMS Speed is derived from the kinetic energy formula and represents the square root of the average of the squares of a set of particular speeds.
The formula to calculate the RMS Speed is: \( v_{rms} = \sqrt{\frac{3RT}{M}} \)where:

• \( R \) is the gas constant, with a value of 8.314 J/(mol K),
  \( T \) represents the temperature in Kelvin,
  \( M \) is the molar mass of the gas.

RMS Speed is a crucial concept because it provides an average speed of gas molecules that accounts for the distribution of speeds present in a sample. It is directly proportional to the temperature and inversely proportional to the molar mass, demonstrating why lighter molecules generally move faster than heavier molecules.

Most Probable Speed

Most Probable Speed is the speed that corresponds to the highest point in the speed distribution curve of gas molecules, essentially indicating the speed at which the highest number of molecules are traveling.
This concept is based on the Maxwell-Boltzmann distribution, which describes the distribution of speeds in a gas. To calculate the most probable speed, you can use the following formula: \( v_{p} = \sqrt{\frac{2RT}{M}} \)where:

• \( R \) is the gas constant,
  \( T \) is the temperature in Kelvin,
  \( M \) is the molar mass of the gas.

Most Probable Speed gives us insight into how temperature and molar mass affect molecular motion. It helps us understand the dynamics within gases under different conditions, particularly emphasizing how rapidly molecules are moving at the most common pace.

Molar Mass

Molar mass is a physical property of substances that tells us the mass of one mole of a chemical element or compound. It is typically expressed in grams per mole (g/mol).
Molar mass plays a vital role in determining molecular speed. According to the formulas used to compute RMS and most probable speeds, we see that molecular speed is inversely proportional to the molar mass.
This means that lighter molecules, with smaller molar masses, generally move faster than heavier ones. For example, in our exercise, CH₄ with the smallest molar mass had the highest molecular speed, demonstrating the significance of molar mass in understanding gas behavior.

Graham's Law of Effusion

Graham's Law of Effusion offers an insightful relation between gas effusion rates and their molar masses. Effusion refers to the process where gas molecules escape through a small hole without collisions among themselves.
Graham's Law states that the rate of effusion of a gas is inversely proportional to the square root of its molar mass, which can be mathematically expressed as:\( \frac{v_1}{v_2} = \sqrt{\frac{M_2}{M_1}} \)Here,

• \( v_1 \) and \( v_2 \) are the effusion rates for gases 1 and 2,
  \( M_1 \) and \( M_2 \) are their respective molar masses.

This law is particularly helpful in comparing different gases' speeds, as it highlights how lighter gases effuse faster than heavier ones, closely relating to concepts of molecular speed and effusion processes.