.MCAD 306000000 Z  docDocument<mcObjectI ( dim_formatCmasslengthtimecharge temperature luminosity substanceNumericalFormat@dii shpRectE@W5`mcDocumentObjectStateJ mcPageModel:????mcHeaderFooter99 |P ComputeEngine=BuiltInsB SerialAnyvalH@@H @@H@HMbP?@H units_classA TextState0 TextStyle/@ ArialNormal/@ArialHeading 1/@ ArialHeading 2/@ ArialHeading 3 /@ ArialParagraph /@ Arial List /@ ArialIndent /@Times New Roman Title /@Times New RomanSubtitle @font_style_list> font_style?  VariablesTimes New Roman?  ConstantsTimes New Roman? TextArial? Greek VariablesSymbol? User 1Arial? User 2 Courier New? User 3System? User 4Script? User 5Roman? User 6Modern? User 7Times New Roman? SymbolsSymbol? Current Selection FontArial? Undefined Font? HeaderArial? FooterArial? Rotated Math FontTimes New Roman$ TextRegion docRegion7shpBoxD5K00 CharacterMapRangeMap,%Thermal Physics PHY474 Computer Lab#5 ChrPropMap(% RangeElem-% ChrPropData) RangeData.lTimes New Roman0,0,128 ParPropMap*%-% ParPropData+EmbedMap"-LinkMap %-%LinkData!@TitleTimes New Roman@DPl`J  BAn important consequence of the second law of thermodynamics is that U(S,V,N) is convex function of S, V and N; F(T,V,N) is concave function of T and convex of V and N; H(S,p,N) is convex of S and N and concave of p; G(T,p,N) is concave of T and p and convex of N; m(T,p) is concave function of T and p. In this lab we study thermodynamic stability through examples from the ideal gas and the blackbody (electromagnetic) radiation. Ideal Gas We use SI units. We take as example the monatomic helium gas: mass of a molecule 4*1.67*10-27Kg; i = 3. The chemical potential as a function of temperature and pressure is given by the Sackur-Tetrode formula which will be derived in the Statistical Physics course. Note that m is a concave function of T and p. ( - )lTimes New Roman0,0,128-)@Symbol0,0,128-$)@Times New Roman0,0,128-)lTimes New Roman0,0,128- )~ 0,0,128!-")l Arial0,0,128#-$)lArial0,0,128!%-&)lArial0,0,128#'-()l Arial0,0,128%)-Z*)lTimes New Roman0,0,128'+-,)LTimes New Roman0,0,128)--4.)lTimes New Roman0,0,128+/-0)@Times New Roman0,0,128-1-A2)lTimes New Roman0,0,128/3-4)@Times New Roman0,0,12815-'6)lTimes New Roman0,0,12837-%8)@Times New Roman0,0,12857*9-:+";- <-=!@NormalArial >eqRegion3@DpnE?tree1 p@@1 ?@A1d@@k.B@B1@@@C1t@B1.381@D1@B@E1t@D10@F1K@D@G1@F23@H3@DnaF@I1 p@J1 @I@K1d@Jm@L1@J@M1@@L@N1t@M4@O1@M1.67@P1@L@Q1t@P10@R1K@P@S1@R27@T3@D{G'P@U1 p@V1 @U@W1d@VR@X1@V8.31@Y3@DvI@Z1 p@[1 @Z@\1d@[h@]1@[@^1t@]6.626@_1@]@`1t@_10@a1K@_@b1@a34@c3@D]@X@d1 p@e1 @d@f1@@e@g1d@f\m@h1p@f@i1 @h@j1d@iT@k1@ip@l1@e@m1@@l@n1K@@m@o1@nR@p1@mT@q1p@l@r1@q@s1@@r@t1@@s@u1@@t@v1@@u@w1t@v5@x1@v2@y1@u@z1d@yln@{1p@y@|1@{T@}1@t@~1d@}ln@1p@}@1@p@1@s@1@@@1t@3@1@2@1@@1d@ln@1p@@1@@1@@@1@@@1t@2@1@\p@1@m@1@@1d@h@1@2@1@r@1@@@1t@5@1@2@1@@1d@ln@1p@@1@k.B@@Dh h^ff@RNote that under standard conditions the chemical potential (j/mole) is negative: (QPR@-P@)@nTimes New Roman0,0,128@-@)@@@-@)@?@@@*R@-R@+"@-@ EmbedData#QEmbedObj$@ EmbedObjPtr%@3@Dh Y@1 p@1@@1@@@1d@\m@1p@@1 @@1t@300@1@@1t@10@1@5@1@@1+@@Serial_DisplayNodeF@1@ R@-R@!@NormalArial @3@D@qJ\@1 p@1 @@1d@i@1@@1t@0@1@19@3@D]@1 p@1 @@1d@j@1@@1t@0@1@19@3@D@)R Z@1 p@1 @@1@@@1d@T@1@i@1@@1t@200@1@@1d@i@1@4@3@D9) [@1 p@1 @@1@@@1d@p@1@j@1@@1t@100000@1@@1d@j@1@100000@3@Dp) `@1 p@1 @@1@@@1d@\m@1 @@1d@i@1@j@1@@1d@\m@1p@@1 @@1@@@1d@T@1@i@1@@1d@p@1@j@3@D75+8@1 p@19@@1d@\m@^@]1 4 3 45 5 1 69 25 0 1 0 1 4 -1 1 0 1 0 1 4 -1 1 0 1 0 1 4 -1 1 30 0 16777215 3 10 2 NO-TITLE@threeDgraphData graphData@ gridderData?@@ xyzTraceDataE@barData!?@@DP#>3P0@:COPYRIGHT @ MIRON KAUFMAN 1997(*@-@+"@- @-@!@NormalArial @@DH>X:6V6VA#The energy is a convex function of entropy and volume. We show this below for the monatomic ideal gas. The energy, the volume and the change in entropy are expressed as dimensionless quantities: u stands for energy/energy0 v stands for volume/volume0 ; Ds stands for (entropy -entropy0)/kB.("S#@-S@)@nTimes New Roman0,0,128@- @)@@Times New Roman0,0,128@@-@)@nTimes New Roman0,0,128@A-A)@NTimes New Roman0,0,128@A-A)@NTimes New Roman0,0,128AA-A)@NTimes New Roman0,0,128AA-A)@NTimes New Roman0,0,128AA-A )@NTimes New Roman0,0,128AA -A )@@Symbol0,0,128AA -A )@@Times New Roman0,0,128A A-A)@@Times New Roman0,0,128A A-A)@@Times New Roman0,0,128AA-A)@@Times New Roman0,0,128AA-A)@@Times New Roman0,0,128AA@*#A-#A+"A- #A-#A!@NormalArial A3@DNfA1 pA1 AA1@AA1dAuA 1pAA!1 A A"1dA!vA#1A!\DsA$1AA%1@A$A&1dA%vA'1A%A(1K@A'A)1A(2A*1A'3A+1A$A,1dA+eA-1A+A.1@A-A/1tA.2A01A.3A11A-\DsA23@DR1dA31 pA41 A3A51@A4A61dA5\DsA71A5iA81A4A91dA8iA:1A8.01A;3@DeA<1 pA=1 A1@A=A?1dA>vA@1A>jAA1A=AB1@AAAC1dABjAD1AB.1AE1AA.1AF3@D`ygAG1 pAH1 AGAI1@AHAJ1dAIuAK1 AIAL1dAKiAM1AKjAN1AHAO1dANuAP1pANAQ1 APAR1@AQAS1dARvAT1ARiAU1AQAV1dAU\DsAW1AUjAX3@D8 (AY1 pAZ19AYA[1dAZu@^@]1 4 3 45 5 1 50 30 0 1 0 1 4 -1 1 0 1 0 1 4 -1 1 0 1 0 1 4 -1 1 30 0 16777215 3 10 2 NO-TITLEA\A]?@A^EA_!?A`@DxE]xXl2jj&Blackbody (Electro-Magnetic) Radiation(&Aa-&Ab)A`lTimes New Roman0,0,128*&Ac-&Ad+"Ae- &Af-&Ag!@NormalArial Ah@Dp1*!<!<@For blackbody radiation: U= aVT4, p = (a/3)T4, S = (4/3)aVT3 , where a is the Stefan constant. Starting with these equations we can derive the following fundamental equations U(V,S) and F(V,T). (=Ai-Aj)AhnTimes New Roman0,0,128Ak-Al)AhNTimes New Roman0,0,128AiAm- An)AhnTimes New Roman0,0,128AkAo-Ap)AhNTimes New Roman0,0,128AmAq-Ar)AhNTimes New Roman0,0,128AoAs-At)AhNTimes New Roman0,0,128AqAu-Av)AhNTimes New Roman0,0,128AsAuAi*Aw-Ax+"Ay- Az-A{!@NormalArial A|3@D<A}1 pA~1 A}A1dA~aA1A~A1tA7.56A1AA1tA10A1KAA1A16A3@DPA1 pA1 AA1@AA1dAFA1pAA1 AA1dAVA1ATA1AA1@AA1@AA1K@AA1AaA1A3A1AVA1AA1dATA1A4A3@DhA1 pA1 AA1@AA1dAUA1pAA1 AA1dAVA1ASA1AA1@AA1@AA1@AA1p@AA1AA1tA3A1A4A1AA1tA4A1A3A1AA1dAaA1AA1K@AA1A1A1A3A1AA1dAVA1AA1K@AA1A1A1A3A1AA1dASA1AA1tA4A1A3A3@DPbzA1 pA1 AA1@AA1dAVA1AiA1AA1p@AA1AA1dAiA1A1A1AA1tA10A1KAA1A3A3@D}A1 pA1 AA1@AA1eATA1AjA1AA1dAjA1A1A3@Dhy{A1 pA1 AA1@AA1dASA1AjA1AA1dAjA1A0.001A3@Dh1(wA1 pA1 AA1@AA1dAUA1 AA1dAiA1AjA1AA1dAUA1pAA1 AA1@AA1dAVA1AiA1AA1dASA1AjA3@DP#9i0xA1 pA1 AA1@AA1dAFA1 AA1dAiA1AjA1AA1dAFA1pAA1 AA1@AA1dAVA1AiB1AB1dBTB1BjB@D[kh:COPYRIGHT @ MIRON KAUFMAN 1997(*B-B+"B- B-B!@NormalArial B 3@DB 1 pB 19B B 1dB F@`@_1 4 3 10 35 1 30 30 0 1 1 1 4 -1 1 0 1 1 1 4 -1 1 0 1 1 1 4 -1 1 21 0 16777215 3 100 2 NO-TITLEB B?@BEB!?B3@D0%B1 pB19BB1dBU@`@_1 4 3 10 35 1 30 30 0 1 1 1 4 -1 1 0 1 1 1 4 -1 1 0 1 1 1 4 -1 1 21 0 16777215 3 100 2 NO-TITLEBB?@BEB!?B@D  *@CNote that F is concave function of T and U is convex function of S.(CB-CB)BnTimes New Roman0,0,128*CB-CB+"B- CB-CB !@NormalArial B!@D(s(:COPYRIGHT @ MIRON KAUFMAN 1997(*B"-B#+"B$- B%-B&!@NormalArial