Question

Show that the rms speed of a gas molecule, defined as \(v_{\mathrm{mm}} \equiv \sqrt{\bar{v}^{2}},\) is given by \(\sqrt{3 k_{\mathrm{B}} T / m}\).

Short answer

The root mean square speed of a gas molecule \(v_{\mathrm{rms}}\) is given by \(\sqrt{3 k_{\mathrm{B}} T / m}\)

Step-by-step solution

Understanding the Concept of rms Speed and Average of Squared Speed

The root mean square (rms) speed of a gas molecule is defined as the square root of the average of the squares of the speeds of the molecules. The kinetic theory of gases provides a relationship between the speed of molecules and the macroscopic properties of a gas, such as temperature and pressure. The rms speed of a gas molecule can be represented as \(v_{\mathrm{rms}} = \sqrt{{v}^{2}}\).

Relate You RMS Speed to the Kinetic Energy

The definition of the rm speed is comparative to the definition of kinetic energy. In kinetic theory of gases, each molecule of an ideal gas has average kinetic energy equivalent to \(\frac{3}{2}\) times Boltzmann’s constant, \(k_{\mathrm{B}}\) times the temperature of gas in Kelvins, \(T\). Hence the average kinetic energy can be mentioned as \( \frac{1}{2} m {v}^{2} = \frac{3}{2} k_{\mathrm{B}} T \). Where \(m\) is the mass of each molecule and \(v\) is the speed of the molecule.

Rearrange the Equation and Solve the rms Speed

Rearrange the equation obtained in step 2 which gives \(v^{2} = 3k_{\mathrm{B}} T / m\). Where \(v = \sqrt{{v}^{2}}\), so \(v_{\mathrm{rms}} = \sqrt{3 k_{\mathrm{B}} T / m}\)

Key concepts

Kinetic Theory of Gases

The kinetic theory of gases is a powerful tool in the field of physics that helps us understand the behavior of gases at the molecular level. According to this theory, gas molecules are considered to be in constant, random motion, colliding with each other and with the walls of their container. This microscopic motion is connected to macroscopic properties like temperature, pressure, and volume.

One of the key ideas of the kinetic theory is that the temperature of a gas is directly proportional to the average kinetic energy of its molecules. When the temperature increases, the molecules move faster, thus increasing their kinetic energy and, consequently, the gas pressure if the volume is kept constant. This relationship is crucial in deriving the root mean square (rms) speed of gas molecules, providing a link between the micro and macro worlds of physics.

The rms speed is particularly useful because it gives us an idea of the speed of a 'typical' molecule in the gas, as opposed to simply knowing the average or most probable speed. By combining this with the mass of a molecule, we can find out a great deal about its kinetic energy and how that relates to the observed properties of the gas.

Boltzmann's Constant

Boltzmann's constant (\(k_{\rm B}\)) is a fundamental constant in physics that appears in various equations relating to the kinetic theory of gases. It acts as a bridge between the micro-scale behavior of individual atoms or molecules and the macro-scale properties of the gas.

Numerically, it's valued at approximately \(1.38 \times 10^{-23} J/K\), where J stands for joules and K for kelvins. Boltzmann's constant essentially tells us how much kinetic energy there is per degree of temperature in each molecule. A key formula in this context is the expression for the average kinetic energy (\(E_{k}\)) of a gas molecule: \(E_{k} = \frac{3}{2} k_{\rm B} T\), where \(T\) is the temperature of the gas in kelvins. This pivotal equation demonstrates how the motions of individual molecules relate to the temperature of a gas, providing a fundamental understanding of thermodynamics.

Average Kinetic Energy

Average kinetic energy is a concept that ties together the microscopic and macroscopic views of gas. It represents the average amount of energy associated with the motion of the molecules in a gas. Kinetic energy is given by the formula \( \frac{1}{2} mv^{2} \), where \(m\) is the mass of a molecule and \(v\) is its velocity.

The average kinetic energy of gas molecules depends solely on the temperature of the gas, regardless of the type of gas or the number of molecules present. This is due to the fact that temperature is a measure of the average kinetic energy per particle in the system. Hence, at a given temperature, all gases have the same average kinetic energy, which is represented by the equation \( \frac{1}{2} m \bar{v^{2}} = \frac{3}{2} k_{\rm B} T \). The bar above \(v^{2}\) signifies that it's the average of the squared velocities of the molecules. Understanding this concept is crucial when solving problems involving the rms speed of gas molecules.

Macroscopic Properties of Gas

Macroscopic properties of gas refer to the physical attributes of a gas that can be observed and measured without the need to look at individual molecules. These include temperature, pressure, volume, and moles of gas, commonly known as the variables in the ideal gas law: \(PV=nRT\).

Temperature, as we have learned, is a measure of the average kinetic energy of the particles in the gas. Pressure is the force exerted by the gas molecules when they collide with the walls of their container. Volume is the space occupied by the gas, and moles (n), a fundamental unit in chemistry, represent the amount of substance or the number of molecules present.

Understanding these macroscopic properties helps in connecting molecular-level phenomena, such as individual molecule speeds and energies, to larger-scale behaviors that we can observe and measure. Being able to calculate quantities like the rms speed of gas molecules - which is derived from kinetic theory and involves Boltzmann's constant - demonstrates the intimate relationship between the microscopic and macroscopic domains in the study of gases.