Problem 85 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

At STP, the order of the RMS speed of molecules of \(\mathrm{H}_{2}, \mathrm{~N}_{2}, \mathrm{O}_{2}\) and HBr gases is (a) \(\mathrm{H}_{2}>\mathrm{N}_{2}>\mathrm{O}_{2}>\mathrm{HBr}\) (b) \(\mathrm{HBr}>\mathrm{O}_{2}>\mathrm{N}_{2}>\mathrm{H}_{2}\) (c) \(\mathrm{HBr}>\mathrm{H}_{2}>\mathrm{O}_{2}>\mathrm{N}_{2}\) (d) \(\mathrm{N}_{2}>\mathrm{O}_{2}>\mathrm{H}_{2}>\mathrm{HBr}\)

Step-by-Step Solution

Answer

\(H_{2} > N_{2} > O_{2} > HBr\)

Step 1: Understand the Concept of RMS Speed

The root mean square (RMS) speed of gas molecules is given by the formula: \(v_{rms} = \sqrt{\frac{3kT}{m}}\), where \(k\) is the Boltzmann constant, \(T\) is the temperature, and \(m\) is the mass of a gas molecule. At standard temperature and pressure (STP), \(T\) is constant, so the RMS speed of the gas molecules is inversely proportional to the square root of the molar mass of the gas.

Step 2: Calculate Molar Mass of each Gas

Find the molar mass of each gas. For \(H_2\), the molar mass is 2 g/mol. For \(N_2\), it is 28 g/mol. For \(O_2\), it is 32 g/mol. For HBr, it is the sum of the molar masses of hydrogen and bromine, which is 1 g/mol + 80 g/mol = 81 g/mol.

Step 3: Compare the RMS Speed Based on Molar Mass

Since the RMS speed is inversely proportional to the square root of the molar mass, we can compare the speeds of the gases by inverting the order of their molar masses. The smaller the molar mass, the higher the RMS speed.

Step 4: Determine the Correct Order of RMS Speeds

Listing the gases in order from the smallest to the largest molar mass: \(H_2\) (2 g/mol), \(N_2\) (28 g/mol), \(O_2\) (32 g/mol), HBr (81 g/mol). Now reverse the order for RMS speeds: \(H_2\) > \(N_2\) > \(O_2\) > HBr.

Key Concepts

Root Mean Square Velocity

The concept of root mean square (RMS) velocity is fundamental in understanding the behavior of gas molecules. It represents the measure of the speed of particles within a gas, providing insight into their kinetic energy. The RMS velocity is calculated using the formula \(v_{rms} = \sqrt{\frac{3kT}{m}}\), where \(k\) is the Boltzmann constant, \(T\) is the absolute temperature, and \(m\) is the mass of an individual gas molecule.

At a given temperature and pressure, all gas molecules have the same kinetic energy. However, because the masses of different gas molecules vary, their speed also varies. Lighter molecules move faster than heavier ones, which is why RMS velocity differs between gases. This concept helps students understand why, under the same conditions, certain gases diffuse more quickly than others and has significant implications in fields like thermodynamics and statistical mechanics.

Molar Mass of Gases

Molar mass is another pivotal concept in the study of gases. It is defined as the mass of one mole of a substance and is typically expressed in grams per mole (g/mol). The molar mass of gases can be determined by adding up the atomic masses of the elements that make up the molecule.

For instance, the molar mass of hydrogen gas \(H_2\) is twice the atomic mass of hydrogen because each molecule consists of two hydrogen atoms. Understanding the molar mass of gases enables students to make predictions about gas behavior, including RMS velocities. This property is instrumental in calculations involving the Ideal Gas Law and other gas-related formulas, which is why grasping the molar mass concept is critical for successfully navigating chemistry problems.

Boltzmann Constant

The Boltzmann constant, denoted by \(k\), is a crucial value in the realm of physics and chemistry, connecting the macroscopic and microscopic worlds. It relates the average kinetic energy of particles in a gas with the temperature of the gas. The constant has units of energy per temperature per particle, typically expressed in joules per kelvin (J/K) or in terms of \(\frac{L\cdot atm}{mol\cdot K}\).

The significance of the Boltzmann constant lies in its role within the RMS velocity formula and its broader relevance in the statistical mechanics of particle systems. It serves as a bridge between macroscopic measurements, like temperature and pressure, and microscopic properties, like the energy of particles.

Standard Temperature and Pressure

When addressing the behavior of gases, it is often necessary to reference a common baseline or standard conditions, known as standard temperature and pressure (STP). This standard is defined as a temperature of 0°C (273.15 K) and a pressure of 1 atmosphere (atm). These conditions are significant because they allow for the comparison of data between different experiments and observations. They serve as the benchmark for calculations and tabulations of gas properties.

At STP, the volume of one mole of an ideal gas is known to be 22.414 liters, and this standard reference makes it easier to discuss and compare the behavior of gases, such as determining their RMS speed. The fact that the temperature is held constant at STP simplifies problems involving the RMS velocity, enabling students to focus on the relationship between the molar mass of the gas and its RMS velocity.