derive an equation for pressure for gas molecules.

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Solution

$C o n s i d e r a n I d e a l g a s e n c l o s e d i n c u b i c a l v e s s e l o f e d g e L, w h e n m o l e c u l e s t r i k e a t o n e p h a s e t h e n c h a n g e i n m o m e n t u m o f m o l e c u l e s, ∆ p = (- m v_{x}) - (- m v_{x}) = - 2 m v_{x} T h e n u m b e r o f c o l l i s i o n o f t h e s e m o l e c u l e s i n u n i t t i m e, n = \frac{1}{∆ t} = \frac{v_{x}}{2 l} w h e r e L i s l e n g t h o f b o x . T h e m o m e n t u m i m p a r t e d t o w a l l p e r u n i t t i m e, ∆ F = n ∆ p = \frac{v_{x}}{2 L} \times 2 m v_{x} = \frac{m}{L} {v_{x}}^{2} I t h a s t h r e e c o m p o n e n t s s o, {v_{x}}^{2} = \frac{1}{3} v^{2} I f t h e r e a r e N n o o f m o l e c u l e s F = \frac{1}{3} \frac{m N}{L} v^{2}, P r e s s u r e, p = \frac{F}{L^{2}} = \frac{1}{3} ρ \frac{v^{2}}{N} = \frac{1}{3} ρ {v^{2}}_{a v g}$

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