hw9_solutions

# hw9_solutions - PHY 251 Fall 2009 homework problem set 9...

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Unformatted text preview: PHY 251 Fall 2009: homework problem set 9, due in the PHY 251 drop box in room A-129 by noon on Friday, Nov. 20. 1. Serway 10.2 Answer: From Serway Eq. 10.8, we know that the Maxwell-Boltzmann velocity distribution is n ( v ) dv = 4 πN V parenleftbigg m 2 πk B T parenrightbigg 3 / 2 v 2 exp[ − mv 2 2 k B T ] dv = B v 2 e- Av 2 dv with A ≡ m 2 k B T and B ≡ 4 πN V parenleftbigg A π parenrightbigg 3 / 2 = 4 N √ πV A 3 / 2 . The most probable speed v mp is at the top of the curve, so we take the derivative and set to zero: d dv B v 2 e- Av 2 dv = B parenleftBig 2 ve- Av 2 + ( − 2 Av ) v 2 e- Av 2 parenrightBig = 2 vBe- Av 2 (1 − Av 2 ) = 0 1 = Av 2 v mp = radicalBigg 1 A = radicalBigg 2 k B T m 2. Serway 10.8. Show the derivations. Answer: We need to rewrite the Maxwell-Boltzmann distribution in terms of kinetic energy K instead of velocity v : K = 1 2 mv 2 → dK = mv dv = √ 2 mK dv → 1 √ 2 mK dK = dv If we substitute these relationships into the expression for n ( v ) dv we get n ( K ) dK = 4 πN V m 3 / 2 2 3 / 2 ( πk B T ) 3 / 2 2 m K exp[ − K k B T ] 1 2 1 / 2 m 1 / 2 √ K dK = 2 πN V 1 ( πk B T ) 3 / 2 √ K exp[ − K k B T ] dK = A √ K exp[ − K k B T ] dK with A ≡ 2 πN V ( πk B T ) 3 / 2 . The peak or K most probable is found by setting the derivative to zero: d dK A √ K exp[ − K k B T ] = 0 A parenleftBigg 1 2 1 √ K exp[ − K k B T ] + √ K exp[ − K k B T ]( − 1 k B T ) parenrightBigg = 0 1 2 √ K = √ K k B T K most probable = k B T 2 1 The mean kinetic energy is found from ( K ) = integraltext K P ( K ) dK . Now n ( K ) gives the num- ber of particles per volume, so dividing by the number of particles per volume gives the probability P ( K ) . Now let’s define x ≡ K/k B T so K = xk B T and dK = k B T dx : ( K ) = integraldisplay ∞ K =0 K 1 N/V 2 πN V π 3 / 2 ( k B T ) 3 / 2 √ K exp[ − K k B T ] dK = integraldisplay ∞ x =0 2 π π 3 / 2 ( k B T ) 3 / 2 ( xk B T ) 3 / 2 exp[ − x ] k B T dx = 2 √ π ( k B T ) integraldisplay ∞ x =0 x 3 / 2 exp[ − x ] dx...
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hw9_solutions - PHY 251 Fall 2009 homework problem set 9...

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