CHEM163B_HW1_solutions_2011 - Winter 2011 Homework Problems...

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Unformatted text preview: Winter 2011 Homework Problems (#1-#5) 1. For an ideal gas PV Chemistry 163B RT (V { Vm { b. V n molar volume) evaluate: a. c. § wP · ¨ ¸ © wV ¹T § wT · ¨ ¸ © wP ¹V § wV · ¨ ¸ © wT ¹ P § wP · § wV · § wT · ¸¨ ¨ ¸¨ ¸ © wV ¹T © wT ¹ P © wP ¹V d. e. One can often get, via the chain rule for differentiation, rigorous results by “canceling w’s” . However the correct result for part d is NOT what one would obtain by just “canceling w’s” in the numerators and denominators in each of the three terms. Why for part d can one NOT § wP · apply the chain rule to get ¨ ¸ 1 ? [NOTE: you may evaluate part d © wP ¹ and e using the specific results from a, b, c for an ideal gas. However the correct result is a more general relationship for any well behaved function z(x,y)] a· § 2. For a Van der Waals gas: ¨ P  2 ¸ V  b RT evaluate: V¹ © § w2 P · § wP · a. ¨ b. ¨ ¸ 2¸ © wV ¹T © wV ¹T § wP · c. ¨ ¸ © wT ¹V ª w § wP · º d. « ¨ ¸» ¬ wT © wV ¹T ¼V ª w § wP · º e. « ¨ ¸» ¬ wV © wT ¹V ¼ T f. How do the results of parts d and e compare? Why is this? 3. Engel & Reid problem P1.3 1 Winter 2011 Chemistry 163B * 4. (optional) The Van der Waals and virial expressions are two commonly used equations of state as approximations to real gas behavior: P RT a 2 (V  b) V Van der Waals P C (T ) º ª C (T ) C2 (T ) C3 (T ) RT « 1   !  n n !» virial 2 3 V V V ¬V ¼ Waals parameters a and b in the following way: C1(T)=1 C2(T)=ba/RT C3(T)=b2 HINT: note that the virial expansion is just a Taylor (Maclaurin) series § 1 · in ¨¸ ©V ¹ k f f P §1· ¦ C k T ¨ V ¸ ¦ Ck T z k f ( z ; T ) RT k 0 © ¹ k0 k a. Show that the first three virial coefficients are related top the Van der zk with coefficient Ck T 1 § dk f · ¨ ¸ k ! © dz k ¹ z 0 b. Why does the Van der Waals “a” only appear in C2(T) ? 5. Limiting cases: a. (required) The residual volume of a gas is defined as: Vres RT · § lim ¨ V  ¸ P o0 P¹ © If a gas has the equation of state PV (1  D P ) what is Vres ? RT * b. (optional) Engel & Reid P1.37 2 5 a. (required) ...
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This note was uploaded on 03/16/2011 for the course CHEM 163B taught by Professor Geneswitkes during the Spring '11 term at University of California, Santa Cruz.

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