ps09soln - PHYSICS 218 SOLUTION TO HW 8 Created: November...

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PHYSICS 218 SOLUTION TO HW 8 Created: November 20, 2004 7:15 pm Last updated: November 21, 2004 1. Schroeder 1.16 (a) The three forces acting on the slab of thickness dz are gravity and the pressure from above and below. To achieve equilibrium they must add to zero. 0= mgρAdz [ P ( z + dz ) P ( z )] A (1) Expanding P(z) in a Taylor series yields P ( z + dz )= P ( z )+ ∂P/∂z × dz = P ( z dP ( z ) and after division by Adz we thus Fnd: dP dz = mgρ (2) (b) The ideal gas law is PV = Nk B T ρ =( mP ) / ( k B T ) inserting gives the barometric equation dP dz = mg k B T P (3) (c) ±or a height independent temperature we use an exponential ansatz P ( z P (0) e λz that yields dP dz = λP (0) e λz = mg k B T P (0) e λz (4) This again gives λ = ( mg ) / ( k B T ) and thus P ( z P (0) e mgz/kT (5) Substituting this function for P ( z ) into the formula for ρ = mP/kT gives us ρ ( z mP (0) kT e mgz/kT = ρ (0) e mgz/kT (6) (d) The numerical value for mg/kT is equal to 1 . 18 × 10 4 /m for an average molecule mass 28 . 8 u and a temperature of 293 K .ThusweFnd Ogden, Utah Leadville, Colorado Mt. Whitney, California Mt. Everest, Nepal/Tibet P / atm 0.84 0.69 0.59 0.35 However we have to bear in mind that these values are approximate. Especially for the Mt. Everest we would Fnd a very di²erent value if we took the temperature decrease into account. 2. Schroeder 1.22 (a) Each molecule hitting the Area A will on average exert a force of 2 m v x , dividing the total force by this amount we Fnd an approximation for the average number of molecules hitting the wall. N = PA 2 m v x t (7) (b) Using ³ v 2 x ´ 1 / 2 as an approximation to v x we use v 2 = v 2 x + v 2 y + v 2 z =3 v 2 x kT/m to Fnd v x q . 1
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PHYSICS 218 SOLUTION TO HW 8 (c) Using the ideal gas law to substitute P and (b) we fnd dN dt = PA 2 m v x = A 2 V s kT m N (8) Again an exponential ansatz N ( t )= N (0) e t/τ will lead us to the result 1 τ N ( t A 2 V s m N ( t ) τ = 2 V A r m (9) (d) Assuming the same data For air as in (1) we fnd the characteristic time τ =6 . 88 s .
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ps09soln - PHYSICS 218 SOLUTION TO HW 8 Created: November...

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