When two atoms collide, some of their kinetic energy may be converted into electronic energy in one or both atoms. If the average kinetic energy is about equal to the energy for some allowed electronic transition, an appreciable number of atoms can absorb enough energy through an inelastic collision to be raised to an excited electronic state. (a) Calculate the average kinetic energy per atom in a gas sample at \(298 \mathrm{~K}\). (b) Calculate the energy difference between the \(n=1\) and \(n=2\) levels in hydrogen. (c) At what temperature is it possible to excite a hydrogen atom from the \(n=1\) level to \(n=2\) level by collision? [The average kinetic energy of 1 mole of an ideal gas is \(\left.\left(\frac{3}{2}\right) R T .\right]\)

Understanding the average kinetic energy of atoms in a gas can provide insight into their thermal motion and potential to cause various physical phenomena. When we talk about average kinetic energy in the context of gas molecules, it is the energy possessed by the molecules due to their motion. The formula to calculate the average kinetic energy of one mole of an ideal gas is \[ K = \left(\frac{3}{2}\right) RT \] where \(R\) is the universal gas constant (8.314 J/mol.K), and \(T\) is the temperature in Kelvin. After finding the average kinetic energy for a mole, to calculate the energy per atom, the result is divided by Avogadro's number (\(6.022 \times 10^{23}\)), since a mole contains that many atoms.

To improve understanding, keep in mind that this calculation assumes ideal gas behavior and ignores interactions between atoms, which is a reasonable approximation at high temperatures and low pressures.

Electrons in atoms occupy specific energy levels. When an electron transitions between these energy levels, it either absorbs or emits energy. These energy level transitions can be quantified using the Bohr model for the hydrogen atom. The energy difference between the \(n=1\) and \(n=2\) levels is given by \[ E = -13.6 \left(\frac{1}{1^2} - \frac{1}{2^2}\right) eV \] because electrons in lower energy states (closer to the nucleus) have more negative energy. In order to compare this energy difference with kinetic energy, it needs to be converted to joules, as the energy is originally given in electronvolts (eV). The conversion is done using the fact that 1 eV = \(1.6 \times 10^{-19}\) J. This information allows us to predict spectral lines in hydrogen and gain understanding of processes such as absorption and emission of light, as well as other quantum mechanical phenomena.

To facilitate comprehension, providing visual aids of energy level diagrams may help to illustrate how these transitions manifest in an atomic spectrum.

The excitation temperature of a hydrogen atom is the temperature at which it has enough thermal kinetic energy to undergo a transition from its ground state (\(n=1\)) to an excited state (\(n=2\)). This transition occurs when the atom collides with another atom or particle, transferring kinetic energy to the electron, enabling it to jump to the higher energy level. The condition for this excitation through collision is that the average kinetic energy of the gas atoms must be comparable to the energy required for the electron transition. As solved in the exercise, equating the average kinetic energy per atom with the energy need for the \(n=1\) to \(n=2\) transition allows for the calculation of the necessary temperature.

Incorporating different methods of visualizing or simulating the process can enhance understanding, such as animations showing a thermal collision leading to electron excitation. Remember, the exact temperature calculated is a theoretical value for an ideal gas and actual conditions may vary due to various physical factors.

An inelastic collision between atoms is a type of collision where part of the kinetic energy is not conserved as kinetic energy but is transformed into other forms of energy, such as electronic energy, resulting in the excitation of electrons. In the context of our exercise, when two hydrogen atoms collide, if the kinetic energy is similar to the energy difference between the electron energy levels, the collision can result in an electron transition from a lower to a higher energy level.

These inelastic collisions are fundamental to understanding how atoms interact and influence each other in a gas. They play a key role in phenomena such as absorption and emission spectra in gases, and the concept also extends to other fields like particle physics and material sciences. Clear and simple explanations, possibly coupled with an interactive model, can reinforce the principle that not all collisions result in the conservation of kinetic energy, highlighting the diversity in atomic interactions.