09_562ln08

09_562ln08 - MIT OpenCourseWare http:/ocw.mit.edu 5.62...

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MIT OpenCourseWare http://ocw.mit.edu 5.62 Physical Chemistry II Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .
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5.62 Lecture #9: CALCULATION OF MACROSCOPIC PROPERTIES FROM MICROSCOPIC ENERGY LEVELS: q trans The macroscopic thermodynamic properties are written in terms of Q. Q is related to the single-molecule partition function q, which is the sum over the molecular energy levels or states. Atoms and molecules have different kinds of states or energy levels. Each type of state makes its contribution, through q, to the macroscopic property under consideration. TRANSLATIONAL CONTRIBUTION TO MACROSCOPIC PROPERTIES V q 2 trans h Single-Molecule Translational Partition Function 2 π mkT 3/2 = Canonical Translational Partition Function N q N trans 1 2 π mkT 2 π mkT Q trans V = = h 2 N! ln Q trans = –ln N ! + N ln V h 2 3 2 N ln 2 π mk h 2 3 = N ln N + N + N ln T + N ln V + 2 (ln N ! = N ln N – N is Stirling’s Approximation) Calculate the translational contribution to the average energy (one of the contributions to U, internal energy) lnQ trans E = kT 2 T N,V 3 N = trans T N,V 2 T E = kT 2 2 3 N T = 2 3 NkT = 2 3 nRT
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5.62 Spring 2008 Lecture 9, Page 2 Average translational energy of N molecules in a gas What about the contribution of translational energy to the heat capacity C V ⎛∂ U T N , V C V trans = E T trans N , V = T 2 3 NkT = 3 Nk = 3 nR 2 2 Recall from 5.60 that C V for an ideal monatomic gas was often observed to be 3 R 2 C V = C V / n = There are no other important places other than translation for an ideal monatomic gas to put internal energy. Calculate translational contribution to pressure : A N,T = kT lnQ V (recall dA = –pdV – SdT + µdN) p = V because A = –kT ln Q 3 2 1 + Nln 2 π mkT h 2 is V dependent) lnQ trans N V = V So p = NkT/V pV = NkT = nRT IDEAL GAS LAW Calculate translational contribution to entropy S = k ln Q + E /T trans = NlnV + ln (only the first term N!
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This note was uploaded on 11/27/2011 for the course CHEM 5.43 taught by Professor Timothyf.jamison during the Spring '07 term at MIT.

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09_562ln08 - MIT OpenCourseWare http:/ocw.mit.edu 5.62...

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