quiz-5-04-07-11-solutions - Quiz 5 Solutions for Physics...

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Quiz 5 Solutions for Physics 176 April 7, 2011 Problems That Require Writing 1. A scientist wants to study the properties of a certain equilibrium solid substance for various values of the temperature T , volume V , and chemical potential μ by placing the substance in a large reservoir for which these three parameters are held constant during a given experiment. (a) (5 points) Find an appropriate thermodynamic potential K ( T,μ,V ) for this experimental setup. Answer: K = U - TS - μN . As discussed in lecture, one can identify or discover a thermodynamic potential by integrating by parts terms in the thermodynamic identity for the thermal energy U until the variables of interest appear as differentials. Here we want a function K ( T,μ,V ) so we proceed as follows: dU = TdS - PdV + μdN (1) = [ d ( TS ) - SdT ] - PdV + [ d ( μN ) - Ndμ ] , (2) which can be rewritten as d ( U - TS - μN ) = dK = - SdT - PdV - Ndμ. (3) Eq. (3) tells us that the quantity K = U - TS - μN is a thermodynamic potential that is a function of the variables T , V , and μ as desired. A challenge: is it possible to find a thermodynamic potential that is a function of the variables T , P , and V ? (b) (5 points) Show how to calculate the entropy S , pressure P , and the number of particles N for this system in terms of K . Answer: The thermodynamic identity for K , Eq. (3) directly tells us that S = - ± ∂K ∂T ² V,μ , P = - ± ∂K ∂V ² T,μ , N = - ± ∂K ∂μ ² T,V . (4) 2. (6 points) A single helium atom with mass m He is placed inside an ideal gas of neon atoms, each with mass m Ne > m He . If the gas is in thermodynamic equilibrium with temperature T , write down in terms of m He , m Ne , and T a mathematical expression for the probability for the He atom to have a speed between 100 m / s and 400 m / s. Note: you do not have to evaluate your expression, just write it down. Answer: prob[100 v 400] = Z 400 100 ³ m He 2 πkT ´ 3 / 2 4 πv 2 e - βm He v 2 / 2 dv, (5) where the mass m He is in kilograms, the speed v in meters per second, and the temperature T is in kelvin if the Boltzmann factor is in the usual SI units. This question tested whether you understand how to identify “what is the system” and “what is the reservoir”. Since the question is asking about statistical properties about the He atom, we choose the single He atom as the system and all Ne atoms as the reservoir. Because the He atom is the system, 1
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the energies E s that appear in the Boltzmann factor and in the partition function Z are the energies of a non-interacting freely moving He atom: E s = E ( v ) = 1 2 m He v 2 , (6) where the different states of the He atom are labeled by the speed v . The Boltzmann factor for a state s with energy E ( v ) is therefore p s = ce - βm He v 2 / 2 , (7) where the constant c is determined by normalization, that the sum of all the probabilities p s must add to 1. But this means that c can only depend on the mass m He since the integral over the probabilities Eq. (7) only involves
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quiz-5-04-07-11-solutions - Quiz 5 Solutions for Physics...

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