Mean Square Speed | Lecture Note - Edubirdie

University:
Santa Fe College
Course:
PHY 2048 | General Physics 1 with Calculus
Academic year:

2024
Views:

304

Pages:

1
Author:

Jabin F.

⇒Mean square speed c 2 =⟨c 2 ⟩= ∫ 0 ∞ p(c)dc ∫ 0 ∞ p(c)c 2 dc 𝑐 2 ‾ = ( ∫ 0 ∞ 𝑒 − 𝑚 𝑐 2 2 𝑘 𝐵 𝑇 𝑐 4 𝑑 𝑐 ) ( ∫ 0 ∞ 𝑒 − 𝑚 𝑐 2 2 𝑘 𝐵 𝑇 𝑐 2 𝑑 𝑐 ) c 2 = (∫ 0 ∞ e − 2k B T mc 2 c 2 dc) (∫ 0 ∞ e − 2k B T mc 2 c 4 dc) 𝑐 2 ‾ = 3 2 ⋅ 𝜋 2 ( 𝑚 2 𝑘 𝐵 𝑇 ) 5 / 2 𝜋 2 ( 𝑚 2 𝑘 𝐵 𝑇 ) 3 / 2 c 2 = 2 π ( 2k B T m ) 3/2 2 3 ⋅ 2 π ( 2k B T m ) 5/2 𝑐 2 ‾ = 3 2 ⋅ 𝑘 𝐵 𝑇 𝑚 c 2 = 2 3 ⋅ m k B T 𝑐 2 ‾ = 3 𝑘 𝐵 𝑇 𝑚 = 3 𝑅 𝑇 𝑀 c 2 = m 3k B T = M 3RT

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