# Speed Distributions When we came up with our formula for the [[Kinetic Theory & Ideal Gases (Thermodynamics)#Root Mean Squared|root mean squared velocity]] we kept most other variables constant, allowing us to treat the system as just 'homogeneous'. However, like many things, a system isn't going to be evenly distributed - even our universe has little 'anisotropies' - irregularities and differences in density here and there. The same can be said about a thermodynamic system, where some particles move faster than others even at the same temperature. **I want you to think: *How can we model the distribution of molecular speeds?*** If you've done some statistics, you may be tempted to say a *normal distribution*. You wouldn't be far off - what we've got here is called a *Maxwell-Boltzmann Distribution*, which is a speed distribution of all the atoms in a sample. Simply put, if we counted up *all* the atoms in one sample, we'd get a certain probability that we can represent on a graph: ![[Pasted image 20231218162216.png]] The probability here represents the chance that we will find a particle that corresponds to a certain velocity. Each graph depends on the temperature of the system, through the equation: $P(v) = 4\pi\left( \frac{M}{2\pi RT} \right)^{\frac{3}{2}}v^2e^{-Mv^2 /2RT}$ This is... a bit of a handful. To derive it, we'll have to dive deep, deep down into statistics. You could almost call this ... [[StatMech - Welcome to Probabilia|statistical mechanics.]] >[!Danger]- Maxwell-Boltzmann Distribution - The Prerequisites >Note that this is statistical mechanics. This means that >[!Danger]- Maxwell-Boltzmann Distribution - The Process (TBA reading handout) # Empirical Data - The Distribution While finding probabilities for a mostly insignificant process is good and all the more practical among you may be itching for a real life use - which we'll deal with right now. The distribution above can be used to find the average, root mean squared and most probable (the tip of the distribution) speeds. Let's start with what we know (the rms) and work our way backwards! We can interpret the graph as being an indicator of the current speed at a certain point. As a result, to find the mean we just integrate over the *entire* expression, taking into account all the velocity values we can throw into the equation: $v^2_{rms} = \int_{0}^{\infty} v^2P(v) \, dv $ This simplifies to $\frac{3kT}{M}$. Look, I am **not** integrating that function up there, alright? Alright, maybe I will... >[!Danger]- Terrible Integral Apply the same logic for the average and most probable speeds! For the average speed, we're dividing only by the curve itself - not by the square. Therefore, the integral we set up is: $v_{avg} = \int_{0}^{\infty} v P(v) \, dv $ $v_{avg} = \sqrt{ \frac{8RT}{\pi M} }$ Finally, we can take the maximum position of the curve to find the most probable speed. A maximum occurs when the first derivative is equal to zero, meaning we can set up an equation as shown: $v_{mp} = \sqrt{ \frac{2RT}{M} }$ # Case Study