5 3 partition function and thermodynamic functions of ideal gases

5.3 Partition function and thermodynamic functions of ideal gases   5.3.1 Partition function of ideal gases  The partition function of an identical system: (1)  For the system of non-interacting particles H=H1+H2+…+HN (2) (3) From (1)(2) and (3), to infer = = (4) 5.3 Partition function and thermodynamic functions of ideal gases   5.3.1 Partition function of ideal gases  As we know, partition function of the ith particle: (5) (4) And (5) => Z =Z1Z2 ZN = (6) We have: Hi =E =U(qi)+K(pi) = (7) 5.3 Partition function and thermodynamic functions of ideal gases   5.3.1 Partition function of ideal gases  Setting (8) (9) To replace (8) and (9) into (7), we have: (10)  Considering the ideal gases including N identical particles freely move in the container of volume V: = 5.3 Partition function and thermodynamic functions of ideal gases   5.3.1 Partition function of ideal gases  We thus have an integral of the form:  This is a standard integral, whose solution can be found in mathematical handbooks Specifically: With: x=p; a=; n=1 5.3 Partition function and thermodynamic functions of ideal gases   =  =(2m 5.3.1 Partition function of ideal gases (10)  The particles of ideal gas can move well inside the volumetric flask, but they can not move beyond the limits of the flask: U(x,y,z)=0 ; =>==V (12)  From (6), (10), (11) and (12), we have:   Z= 5.3 Partition function and thermodynamic functions of ideal gases • 5.3.2 The thermodynamic potentials  Free energy =( + N) =( + )  Where lnN! NlnN −N= N(lnN −lne) = N ( by STIRLING formula)   F =( ) 5.3 Partition function and thermodynamic functions of ideal gases • 5.3.2 The thermodynamic potentials  Pressure = = N.()    = 5.3 Partition function and thermodynamic functions of ideal gases •F=( )  5.3.2 The thermodynamic potentials  Entropy = =(+T =( +)    S =( +) 5.3 Partition function and thermodynamic functions of ideal gases •  Internal energy 5.3.2 The thermodynamic potentials E =( - + +) =( - -1 - - ln()+ +)    E= ... ) 5. 3 Partition function and thermodynamic functions of ideal gases • 5. 3. 2 The thermodynamic potentials  Pressure = = N.()    = 5. 3 Partition function and thermodynamic functions of ideal gases. . .5. 3 Partition function and thermodynamic functions of ideal gases   5. 3. 1 Partition function of ideal gases  As we know, partition function of the ith particle: (5) (4) And (5) => Z... ideal gases including N identical particles freely move in the container of volume V: = 5. 3 Partition function and thermodynamic functions of ideal gases   5. 3. 1 Partition function of ideal gases