The root mean square velocity of hydrogen is \(\sqrt{5}\) times than that of nitrogen. If \(T\) is the temperature of the gas, then : (a) \(T_{\mathrm{H}_{2}}=T_{\mathrm{N}_{2}}\) (b) \(T_{\mathrm{H}_{2}}>T_{\mathrm{N}_{2}}\) (c) \(T_{\mathrm{H}_{2}}

Physical chemistry is the branch of chemistry concerned with the underlying principles that govern chemical phenomena. It meshes the macroscopic world that we observe with the microscopic behavior of atoms and molecules to explain why materials behave the way they do. One of the main topics within physical chemistry is the study of gases and their properties, which includes understanding concepts such as temperature, pressure, volume, and the molecular speeds that are associated with different gas particles.

One of the key quantitative measures used in this field is the root mean square velocity, which is a way to describe the average speed of molecules in a gas. This concept is crucial because it directly relates to the kinetic energy of the gas and therefore its temperature. Understanding how temperature affects the behavior of gases, and how this can be quantified, is an essential aspect of physical chemistry.

The Kinetic Molecular Theory (KMT) of gases offers a microscopic explanation of the properties of gases, including pressure, temperature, and volume. According to KMT, a gas is composed of a large number of particles that are in constant, random motion and that these particles frequently collide with each other and the walls of their container.

Root Mean Square Velocity

A fundamental concept derived from KMT is the root mean square velocity, which represents the average velocity of the molecules in a gas. This is particularly important as it can be used to determine how the molecules' speeds change with temperature. It is derived as a function of both the gas's temperature and the molar mass of its molecules, giving us a direct link between the microscopic and macroscopic properties of a gas. The given problem illustrates how changes in molecular speed can hint at changes in these other properties, such as temperature.

The Boltzmann Constant, denoted as \( k \), is a fundamental constant in physical chemistry that bridges the gap between macroscopic and microscopic physics. It appears in various key equations that describe the behavior of particles in thermodynamic systems. The Boltzmann Constant relates the average kinetic energy of particles in a gas to the absolute temperature of the gas.

In the context of the root mean square velocity, it is crucial because it links the average kinetic energy to a measureable property, the temperature. The use of the Boltzmann Constant in the formula for root mean square velocity provides a direct relationship between temperature and molecular motion within a gas which is pivotal for solving problems related to thermodynamics and molecular behavior.

In chemistry, the molar mass is the mass of one mole of a substance, typically expressed in grams per mole (g/mol). This value is fundamental when dealing with chemical amounts, reactions, and when predicting the physical behavior of chemical compounds.

When studying gases, the molar mass becomes particularly important in the context of the root mean square velocity. Since lighter molecules move faster at a given temperature than heavier molecules, the molar mass inversely affects the root mean square velocity of a gas. Understanding the molar mass of molecules allows us to predict their speed and how it will vary with changes in temperature. This concept is exemplified in the given exercise, where it's utilized to compare the temperatures of hydrogen and nitrogen based on their different molar masses.