What is the degree of freedom in kinetic gas theory? How can I determine how much degree of freedom some molecule has?
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A degree of freedom is a parameter that can have any value it wants (if you want to think it can be any element of a field). This means, there is no restriction for this number to be fixed. In kinetic gas theory, a degree of freedom usually first appears in the energy of a monoatomic molecule as being $E = \frac{3}{2}T$ and a diatomic molecule being $E= \frac{5}{2}T $ and the explanation goes as follows:
-The monoatomic molecule can move in x,y or z thus there are 3 degrees of freedom
-The diatomic molecule can move in x,y, or z (the center of mass) but also can rotate (as you may now, the sphere can be parametrized with 2 parameters) thus the total number of degrees of freedom is now 5.
Determining how many degrees of freedom some molecule has is not an easy task, this in the sense that the more energy you input to the system, the more you can understand it. Imagine you are one a 1D universe and there are 2 particles joined by a very strong spring. At first sight, you might say there is only one degree of freedom since it can only move in one dimension, but as you give more and more energy to the system it can start oscillating (due to the spring) so new degrees of freedom arose. This is the idea of entropy, to quantify how bad we know our system. If we know all the degrees of freedom of the system, it will have low entropy (for us), but if there are "hidden" degrees of freedom, it will have high entropy.
In order to determine the total number of degrees of freedom, you need to know your system, and unless you have an explicit description of it, you can count them. So let us say, we have 2 diatomic particles. Each we know it can be on any x, y, or z and can rotate in 2 different directions, so in total, we will have 10 degrees of freedom. I hope this answers your question.
