Chapter 39: Problem 35
A large number of neon atoms are in thermal equilibrium. What is the ratio of the number of atoms in a 5\(s\) state to the number in a 3\(p\) state at (a) 300 K; (b) 600 K; (c) 1200 K? The energies of these states, relative to the ground state, are E\(_{5s}\) = 20.66 eV and E\(_{3p}\) = 18.70 eV. (d) At any of these temperatures, the rate at which a neon gas will spontaneously emit 632.8-nm radiation is quite low. Explain why.
Short Answer
Step by step solution
Understand the concept
Apply the Boltzmann distribution
Calculate for 300 K
Calculate for 600 K
Calculate for 1200 K
Discuss spontaneous emission at 632.8 nm
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Neon atoms
In physics, neon atoms are often studied in terms of their atomic structure and energy states. Each neon atom consists of a nucleus surrounded by electrons stacked in various energy levels or orbitals. These energy levels are central to how neon atoms interact with radiation and heat.
Because neon atoms have clearly defined energy levels, they are excellent subjects for studying how atoms transition between these energy states under different conditions, such as varying temperatures.
Energy states
Each energy state can be uniquely identified by its energy value, for instance, given as \(E_{5s}\) and \(E_{3p}\) in the problem. Here, the 5s and 3p labels denote specific electron configurations in these higher energy levels. Each of these states possess a specific energy difference compared to a reference point, often the ground state.
These energy differences are crucial for predicting how many atoms occupy each state at a given temperature. The population of atoms in these energy states can be determined using statistical methods, such as the Boltzmann distribution, which is a key concept in statistical mechanics.
Thermal equilibrium
When neon atoms are in thermal equilibrium, their energy state populations are described by the Boltzmann distribution. This distribution shows that more atoms will occupy the lower energy states than higher ones, especially at lower temperatures. The ratio of atoms in different energy states at thermal equilibrium becomes a focus of study because it reveals important insights into atomic behavior and is dictated by variables such as temperature and energy differences between states.
- The lower the temperature, the fewer atoms will be in higher energy states.
- As temperature increases, more atoms can jump to higher energy levels.
Spontaneous emission
For neon, when an electron from a higher energy state such as \(5s\) spontaneously drops to a lower state like \(3p\), light is emitted. The study of this behavior helps in understanding the spectra of gases and is used in applications like gas discharge lamps.
However, as noted in the exercise, spontaneous emission rate is quite low at 632.8 nm. This low rate occurs because:
- There are fewer atoms in the higher \(5s\) state, especially at lower temperatures, reducing potential emission events.
- Transition rates for spontaneous processes are inherently slower compared to stimulated processes.
- Low thermal energies in typical conditions further decrease the likelihood of spontaneous emissions.