Problem Set 5

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Oct 30, 2023

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Chemistry 120B, Fall 2023 Problem Set 5 Due Friday, Oct. 6 1. (35 points) The classical partition function for an imperfect gas comprising N atoms is given approximately by 2 0 2 2 3 1 ( ) , ! N cl N V N Q Q Q B T N V   = = −      where 1/2 (2 ) B h mk T   = and ( ) 2 ( )/ 2 2 0 1 ( ) 4 1 2 u R kT B T R e R dR   − = − −  is the second virial coefficient. a) Derive a general expression for the energy 2 , ln cl N V Q E kT T    =      of the imperfect gas in terms of B 2 (T) and/or dB 2 (T)/dT. You may use the same approximation we used in class when finding ln(Q 2 ). b) The Lennard-Jones potential is given by 12 6 ( ) 4 u R R R          = −               . For this potential, 3 3 2 2 16 ( ) 3 9 B B T k T   = − . Find an expression for 0 / E E  , the fractional correction to the ideal gas energy E 0 , where  E , which will depend on  and  , is the contribution from the Lennard- Jones interaction. c) For a pair of Ar atoms, the Lennard-Jones parameters are  =3.61 Å and  =83 cm -1 . Plot this potential, with values for  , R e (the value of R at the potential energy minimum), and  indicated on the appropriate axes. Calculate B 2 (T) and 0 / E E  for Ar gas at P=1.0 bar and T=298 K. Explain why these values are positive and/or negative. 2. (35 points) We have previously shown that for an ideal gas, the Helmholtz free energy 3 ln ln Ve F kT Q NkT N   = − = −      . (a) Using the ideal gas law PV=NkT, show that the Gibbs free energy G=F+PV is given by 3 ln kT G NkT P   = −      . Find the chemical potential  under constant volume and constant pressure , i.e. , V T F N     =      and , T P G N     =      . Are the results the same or different?

(b) We can also define the chemical potential in terms of derivatives of the energy E and enthalpy H. For a one component system, these are , V S E N     =      and , P S H N     =      . Evaluate these expressions for an ideal gas and compare to those in part (a). Hint: Use the cyclic rule (Cooksy, p.16) as a first step. 3. (25 points) The charge-dipole interaction between Ne + and HCl separated by distance r is given by 2 cos ( ) q V r r   = , where q is the charge on the Ne + ,  is the dipole moment of the HCl, and  is the angle between the HCl dipole moment and the vector r between the Ne + ion and the HCl center of mass. This expression holds for a particular orientation of the HCl molecule. Set up an expression for ( ) ( ) V r V r = , i.e. the charge-dipole potential averaged over all orientations (  ,  ) at temperature T, and evaluate this expression. Here are some possibly useful formulae for the evaluation: 1 1 1 1 2 sinh( ) 2 sinh( ) 2cosh( ) 1 coth( ) 3 ax ax e dx a a ax e dx a a a a a a − − − − =  = −  +  

Problem Set 5

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