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Pre-Lecture Homework
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4.1 Pre-Lecture Homework
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4.1.1 Boltzmann Distribution
Even before we had the statistical interpretation of quantum mechanics, statistics was applied to fundamental science to account for observations. Ludwig Boltzmann and other founders of statistical mechanics used methods of probability theory to bridge the microscopic world and macroscopic observations such as temperature, work, heat and chemical processes.
Say we have a macroscopic system at some temperature T. Suppose there exists a variety of possible "states" (state 1, state 2, etc), each with its characteristic energy (E_{1}, E_{2}, etc.). The probability of occurence of a state is related to the state's energy via the Boltzmann distribution:
\text{Probability of state } i \text{ occurring }=p_{i}\propto e^{-E_{i}/kT}
where k=1.38\times10^{23}\text{JK}^{-1} is the Boltzmann constant.
Try it yourself!
Have you heard of the phrase "particles prefer lower energy states"? Explain this using Boltzmann distribution.
Try it yourself!
In the atomic spectrum experiment (hope that you have started doing it!), you will see that the intensities of each spectral line is different. Boltzmann distribution partly explains this. How?
📖 Read!
Read up more on Boltzmann distribution and find examples of how it is used in other fields.
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4.1.2 Clouds in outer space
The space between stars is not empty. It is dotted with matter, at some places denser than others. As they exist in the regions between and around stars, they are known as the interstellar medium (ISM). The ISM can be made of clouds of hydrogen and helium gas, as well as "dust grains" that contain heavier elements. Interactions between matter and light within the ISM give rise to beautiful astronomical objects called nebulae.
Try it yourself!
Why is an emission nebula reddish in colour, and a dark nebula dark?