PhysicsChapter 2: Statistical Mechanics and Thermodynamics
Mean-Square and Rms Speed
The **mean-square speed** is the average of the squares of molecular speeds: **⟨c²⟩** The **root-mean-square speed** is: c_rms = √⟨c²⟩
Definition
The mean-square speed is the average of the squares of molecular speeds: ⟨c²⟩ The root-mean-square speed is: crms = √⟨c²⟩
Formula
crms =
Important Distinction
Gas molecules do not all move at the same speed. Suppose molecular speeds are: `c₁, c₂, c₃ ... cN` Mean-square speed is: ⟨c²⟩ = c₁²+c₂²+...+cN²N RMS speed is: crms = √[c₁²+c₂²+...+cN²N] RMS speed is not generally the same as ordinary average speed. Because pressure depends on squared velocity components, RMS speed appears naturally in kinetic theory. From: **p=(13)ρ⟨c²⟩ and: crms²=⟨c²⟩ we get: p=(13)ρcrms² Therefore: crms=√(3p/ρ)**
Formula / Key Relation
**crms = √⟨c²⟩ crms = √(3p/ρ)**
Exam preparation
Exam Focus
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Review the key facts, formulas and checks before you move on.Mean-Square and Rms Speed
Sticky Notes · Quick Revision
RMS:
- 1. Square
- 2. Mean
- 3. Root
60-second revision
- RMS is directly linked to molecular kinetic energy and pressure.
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