3 Dimension Tutorial Notes | Santa Fe College - Edubirdie

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

2024
Views:

266

Pages:

3
Author:

Jabin F.

Consider an ideal 3-D gas consisting of N non-interacting molecules filled in a 3-D container of volume V at absolute temp T. E = p²/2m = 1/2 mv² = 1/2 mc² m = mass of molecule c = speed of a molecule c ≠ speed of light in vacuum => The probability of having speed of a molecule lies in range c to c+dc P(c)dc ∝ e^(-βE) 4π²c² dc P(c)dc = A e^((-mc²)/(2kBT)) 4π²c² dc ∫[0,∞] P(c)dc = 1 A 4π ∫[0,∞] e^((-mc²)/(2kBT)) c² dc = 1 A 4π √(3/2)/(2(m/(2kBT))^(3/2)) = 1 A 4π (√π/2)/(2(m/(2kBT))^(3/2)) = 1 A = (m/(2πkBT))^(3/2) P(c)dc = (m/(2πkBT))^(3/2) e^((-mc²)/(2kBT)) 4πc² dc The probability of having sharp value of speed c by a molecule P(c) = (m/(2πkBT))^(3/2) e^((-mc²)/(2kBT)) 4πc² => P(c) = 4π (m/(2πkBT))^(3/2) e^((-mc²)/(2kBT)) c² (having speed) The no. of molecules lies in the range of speed c to c+dc n(c) dc = N P(c) dc = 4πN (m/(2πkBT))^(3/2) e^((-mc²)/(2kBT)) c² dc ∫₀ⁿ n(c) dc = N ∫₀ⁿ P(c) dc = N

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