Question

An atom of germanium (diameter \(=246 \mathrm{pm}\) ) escapes from a furnace \((T=4220 \mathrm{~K})\) with the root-mean-square speed into a chamber containing atoms of cold argon (diameter = \(300 \mathrm{pm}\) ) at a density of \(4.13 \times 10^{19}\) atoms/cm \(^{3}\). ( \(a\) ) What is the speed of the germanium atom? ( \(b\) ) If the germanium atom and an argon atom collide, what is the closest distance between their centers, considering each as spherical? ( \(c\) ) Find the initial collision frequency experienced by the germanium atom. -

Short answer

The speed of the germanium atom can be calculated according to the method shown in Step 1. The closest distance lies between the distances calculated in Step 2. The collision frequency is determined based on the formula mentioned and outlined in Step 3.

Step-by-step solution

Root Mean Square Speed of Germanium Atom

The root mean square speed of an atom can be given by the formula \[v_{rms}=\sqrt{\frac{3kT}{m}}\] where: \(k\) is Boltzmann constant = \(1.38 \times 10^{-23} \, \mathrm{J/K}\), \(T\) is the temperature in Kelvin = 4220 K, and \(m\) is the mass of one germanium atom which can be calculated by \[m=\frac{Molar\, Mass\, of\, Germanium}{Avogadro's\, Number}\].

Closest Distance Between a Germanium Atom and Argon Atom

Atoms are treated as hard spheres in the kinetic theory of gases. The closest distance between the centers of colliding atoms when they undergo a collision corresponds to the sum of their radii. Considering diameters of germanium and argon atoms to be 246 pm and 300 pm respectively, the distance can be calculated by: \[Distance = Radius\, of\, Germanium\, Atom + Radius\, of\, Argon\, Atom = \frac{Diameter\, of\, Germanium}{2} + \frac{Diameter\, of\, Argon}{2}\].

Collision Frequency of Germanium Atom

The initial collision frequency can be calculated by the formula: \[z = nv_{rms} \sigma_z\] where: \(n\) = density of argon = \(4.13 \times 10^{19} cm^{-3}\), \(σ_z\) = collision cross section which can be calculated by \[σ_z = π* (Diameter\, of\, Germanium\, Atom +Diameter\, of\, Argon\, Atom)^2\].

Key concepts

Kinetic Theory of Gases

The kinetic theory of gases is a fundamental principle that describes the behavior of gases in terms of the motion of their constituent atoms or molecules. According to this theory, gas particles are in constant, random motion and their collisions with each other and with the container walls result in the observed pressure. The theory assumes that these collisions are perfectly elastic, meaning no energy is lost, and the particles themselves are considered to be point masses with no volume.

Under the kinetic theory, temperature is directly related to the average kinetic energy of the gas particles. Higher temperatures mean greater average kinetic energy and ultimately higher velocities. This theory helps in understanding properties like pressure, temperature, and volume of a gas, and explains how they are all interconnected. It also provides insight into the root mean square speed, which measures the average speed of the gas particles, and is crucial for solving problems like the one concerning the germanium atom's behavior in a chamber.

Collision Frequency

Collision frequency is a term that refers to the number of collisions a single particle experiences per unit time. In a gas, particles are constantly moving and colliding with one another. The frequency of these collisions is influenced by factors such as the size and speed of the particles and the density of the gas.

In our scenario, the germanium atom escapes with a certain speed and moves into a chamber filled with argon atoms. Collision frequency would help in determining how often the germanium atom is likely to collide with the argon atoms in this chamber. For this purpose, physicists use mathematical models to calculate this frequency, ensuring they have a solid understanding of the dynamics within gaseous systems. This concept is vital in fields such as chemical kinetics and the study of reaction mechanisms where collision frequencies affect reaction rates.

Boltzmann Constant

The Boltzmann constant (\(k\)) is a fundamental constant that appears in several key equations of physics and chemistry, particularly in the kinetic theory of gases. It relates the average kinetic energy of particles in a gas with the temperature of the gas in the equation \(E_{k} = \frac{3}{2}kT\).

The value of the Boltzmann constant is approximately \(1.38 \times 10^{-23} \, \mathrm{J/K}\) and serves as a bridge between macroscopic and microscopic physics. It is also involved in the definition of entropy and the distribution of energies among the particles in a system (Boltzmann distribution). In the context of our exercise, the Boltzmann constant is used to calculate the root mean square speed of the germanium atom, giving us an understanding of how fast it is moving at a given temperature.

Avogadro's Number

Avogadro's number is another fundamental constant of nature, defining the number of atoms, ions, or molecules in one mole of a substance. It is approximately equal to \(6.022 \times 10^{23}\) and is essential for converting from a microscopic scale to a macroscopic one. For example, when determining the mass of a single atom of germanium, we take the molar mass (the mass of one mole of atoms) and divide it by Avogadro's number.

Understanding Avogadro's number is crucial when dealing with physical constants and converting between atoms and moles. For our exercise, this number allows us to calculate the mass of a germanium atom in kilograms, which then can be used in the equation for root mean square speed. Without Avogadro's number, this kind of conversion would not be possible, showcasing the importance of this constant in solving chemistry and physics problems related to the amount of substance.