.MCAD 306000000 Z  docDocument<mcObjectI (d2_graph_format graphData axisFormatLLtrace2D      dim_formatCmasslengthtimecharge temperature luminosity substance NumericalFormat@dii shpRectEe@mcDocumentObjectStateJ mcPageModel:????mcHeaderFooter99 ComputeEngine=BuiltInsB2 SerialAnyvalH@1@H @/@H0@HMbP?,@H units_classA TextState0 TextStyle/@ Times New Roman0,0,128Normalfont_style_list> font_style?  VariablesTimes New Roman?  ConstantsTimes New Roman? TextTimes New Roman? 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 Roman8 TextRegion docRegion7shpBoxD7=0/8/8 CharacterMapRangeMap,CCOPYRIGHT MIRON KAUFMAN 1997 THERMAL PHYSICS COMPUTER LAB #2 We compute the work and the heat for each step of the Stirling cycle: two isochores, two isotherms. The efficiency of a heat engine using the Stirling cycle will also be calculated. The thermodynamic system is an ideal van der Waals fluid. The equation of state is: (p (p + a/v2)(v - b) = RT. The energy is: u = i/2*RT - a/v. The a terms originate in the long-range inter-molecular attraction. The b term represents the molar volume corresponding to close packing of the molecules (i.e. under high pressures). i is number of degrees of freedom per molecule: 3 - monatomic gas, 5 -diatomic gas, 6-polyatomic gas. This sort of engine was used in the past to drive printing presses and other fixed machines. Nowadays it is considered for use in power plants using solar energy. The values for i, a and b below are for nitrogen N2. If you want a different gas then you have to change those values.  ChrPropMap($ RangeElem-  ChrPropData) RangeData. - ) -$ )|Britannic Bold -)}Britannic Bold -)}-)-){-/)-)_-5)-)w-E) -!)w"-p#) $-%)w"&-e')$(-))&*-3+)(,--)_*.-D/),.  ParPropMap*S0-1 ParPropData+2-'3+04-k5+26-7+48-x9+680EmbedMap":-LinkMap ;-<LinkData!@NormalTimes New Roman =@DUu`  @nWe can use the symbolic processor to evaluate the work done by the gas in an isothermal process. dW = - pdV. (n*n>-n?+"@@- n@A-n@B!@NormalTimes New Roman @CeqRegion3@D{7@Dtree1 p@E1?@D@F1K@@E@G1%@F@H1@@G@I1d@HV.1@J1@HV.2@K1@G@L1d@KV@M1@K@N1@@M@O1@@N@P1@@O@Q1d@PN@R1@PR@S1@OT@T1@N@U1d@TV@V1@T@W1d@VN@X1@Vb@Y1@M@Z1@@Y@[1d@Za@\1@Z@]1d@\N@^1@\2@_1@Y@`1d@_V@a1@_2@b1@E@c@DT@DD1Next we will do the the computations numerically.(1*1@d-1@e+"@f- 1@g-1@h!@NormalTimes New Roman @i3@D{@j1 p@k1 @j@l1d@kR@m1@k@n1t@m8.31451@o1@m@p1d@ojoule@q1@o@r1d@qK@s1@qmole@t3@D @u1 p@v1 @u@w1d@vN@x1@v@y1t@x1.5@z1@xmole@{3@D!sI8=@|1 p@}1 @|@~1d@}a@1@}@1@@@1t@0.136@1@Pa@1@@1@@@1d@m@1@6@1@@1d@mole@1@2@3@D!E8C@1 p@1 @@1d@b@1@@1@@@1t@38.5@1@@1t@10@1K@@1@6@1@@1@@@1d@m@1@3@1@mole@3@D.= 8D@1 p@1 @@1d@i@1@5@3@DM|q7`@1 p@1 @@1@@@1d@p@1p@@1 @@1d@v@1@T@1@@1@@@1@@@1d@R@1@T@1@@1d@v@1@b@1@@1d@a@1@@1d@v@1@2@3@DM3m`@1 p@1 @@1@@@1d@u@1p@@1 @@1d@v@1@T@1@@1@@@1@@@1@@@1d@i@1@2@1@R@1@T@1@@1d@a@1@v@@D<84]4]A=Before proceeding with the cycle calculation we will learn to use the Mathcad root function which solves numerically a given equation. The root function requires a guess of the root. Let us find the molar volume of the gas under atmospheric pressure p = 1atm = 1.01*105Pa and at room temperature T = 300K. Guess: (<N=@-N@)@@-@)@mTimes New Roman0,0,128@@-@)@@@-@)@_@@--@)@@@-@)@?@@@*=@-=@+"@-@ EmbedData#<P$EmbedObj$@ EmbedObjPtr%@3@D@1 p@1 @@1d@v@1@@1t@0.01@1@@1@@@1d@m@1@3@1@mole =@-=@!@NormalTimes New Roman @3@DY@1 p@1@@1@@@1d@root@1p@@1 @@1@@@1@@@1d@p@1p@@1 @@1d@v@1@@1t@300@1@K@1@@1@@@1t@1.01@1@@1t@10@1@5@1@Pa@1@vA1@A1+@ASerial_DisplayNodeFA1AA@D8@@So the molar volume under standard conditions is 25liters/mole. (@*@A-@A+"A- @A-@A!@NormalTimes New Roman A @D-(8COPYRIGHT MIRON KAUFMAN 1997(*A -A +"A - A -A!@NormalTimes New Roman A3@DIm`A1 pA1 AA1dAvA1AA1 @AA1@AA1@AA1tA5A1AA1tA10A1KAA1A3A1AA1@AA1dAmA1A3A 1AmoleA!1AA"1@A!A#1tA"6A$1A"A%1tA$10A&1KA$A'1A&3A(1A!A)1@A(A*1dA)mA+1A)3A,1A(moleA-1AA.1@A-A/1tA.40A01A.A11tA010A21KA0A31A23A41A-A51@A4A61dA5mA71A53A81A4moleA9@DFLH> > @Next we graph the cycle: 1 --- 2 isothermal compression; 2 --- 3 isochoric heating; 3 --- 4 isothermal expansion; 4 --- 1 isochoric cooling. (*A:-A;+"A<- A=-A>!@NormalTimes New Roman A?3@DO A@1 pAA1A@AB1@AAAC1@ABAD1@ACAE1BADAF1dAE 6.648823AG1AEAH1dAG10AI1AG5AJ1ADAK1dAJ4.15276AL1AJAM1dAL10AN1AL4AO1ACAP1dAO _n_u_l_l_AQ1AO _n_u_l_l_AR1 ABAS1@ARAT1@ASAU1dATpAV1pATAW1 AVAX1dAWvAY1AWAZ1tAY200A[1AYKA\1ASPaA]1ARA^1@A]A_1dA^pA`1pA^Aa1 A`Ab1dAavAc1AaAd1tAc400Ae1AcKAf1A]PaAg1AAAh1@AgAi1@AhAj1fAi0.04Ak1AiAl1dAk5Am1AkAn1dAm10Ao1KAmAp1Ao3Aq1AhAr1tAq0.015As1Aq0.025At1 AgAu1@AtAv1dAuvAw1pAuAx1AwAy1@AxAz1dAymA{1Ay3A|1AxmoleA}1AtA~1dA}vA1pA}A1AA1@AA1dAmA1A3A1AmoleAA% \v(cubic meters)\ p(Pascals) The Stirling cycle in v,p plane.     A@D-=8"@@PProcess 1 to 2 is isothermal compression while in contact with cold reservoir. (P*PA-PA+"A- PA-PA!@NormalTimes New Roman A3@DYN}.pA1 pA1 AA1dA\DU.12A1AA1dANA1pAA1AA1@AA1dAuA1pAA1 AA1@AA1tA0.015A1AA1@AA1dAmA1A3A1AmoleA1AA1tA200A1AKA1AA1dAuA1pAA1 AA1@AA1tA0.025A1AA1@AA1dAmA1A3A1AmoleA1AA1tA200A1AKA3@Dx`zp~A1 pA1AA1dA\DU.12A1AA1+@A@FA1A _n_u_l_l_A3@D+|A1 pA1 AA1dAW.12A1KAA1%AA1@AA1@AA1tA0.025A1AA1@AA1dAmA1A3A1AmoleA1AA1tA0.015A1AA1@AA1dAmA1A3A1AmoleA1AA1dAvA1AA1@AA1dApA1pAA1 AA1dAvA1AA1tA200A1AKA1ANA3@DBA1 pA1AA1dAW.12A1AA1+@A@FA1A _n_u_l_l_A3@D |"(}A1 pA1 AA1dAQ.12A1AA1dA\DU.12A1AW.12A3@DF) A1 pA1AA1dAQ.12A1AA1+@A@FA1A _n_u_l_l_A@DMHmXH  COPYRIGHT MIRON KAUFMAN 1997 (*A-A+"A- A-A!@NormalTimes New Roman A@D}5#@--1Process 2 to 3 is isochoric raise in temperature.(1*1A-1A+"A- 1A-1A!@NormalTimes New Roman A3@DN.A1 pA1 AA1dA\DU.23A1AA1dANA1pAB1AB1@BB1dBuB1pBB1 BB1@BB1tB0.015B1BB1@BB 1dBmB 1B3B 1BmoleB 1BB 1tB 400B1B KB1BB1dBuB1pBB1 BB1@BB1tB0.015B1BB1@BB1dBmB1B3B1BmoleB1BB1tB200B1BKB3@D`B1 pB1BB 1dB\DU.23B!1BB"1+@B!@FB#1B! _n_u_l_l_B$3@D[+B%1 pB&1 B%B'1dB&W.23B(1B&B)1tB(0B*1B(jouleB+3@D`B,1 pB-1B,B.1dB-W.23B/1B-B01+@B/@FB11B/ _n_u_l_l_B23@D|(B31 pB41 B3B51dB4Q.23B61B4B71dB6\DU.23B81B6W.23B93@DX wB:1 pB;1B:B<1dB;Q.23B=1B;B>1+@B=@FB?1B= _n_u_l_l_B@@D $@) Process 3 to 4 is isothermal expansion. ()*)BA-)BB+"BC- )BD-)BE!@NormalTimes New Roman BF3@D!NE.8BG1 pBH1 BGBI1dBH\DU.34BJ1BHBK1dBJNBL1pBJBM1BLBN1@BMBO1dBNuBP1pBNBQ1 BPBR1@BQBS1tBR0.025BT1BRBU1@BTBV1dBUmBW1BU3BX1BTmoleBY1BQBZ1tBY400B[1BYKB\1BMB]1dB\uB^1pB\B_1 B^B`1@B_Ba1tB`0.015Bb1B`Bc1@BbBd1dBcmBe1Bc3Bf1BbmoleBg1B_Bh1tBg400Bi1BgKBj3@Dp( B8Bk1 pBl1BkBm1dBl\DU.34Bn1BlBo1+@Bn@FBp1Bn _n_u_l_l_Bq3@DP+Br1 pBs1 BrBt1dBsW.34Bu1KBsBv1%BuBw1@BvBx1@BwBy1tBx0.015Bz1BxB{1@BzB|1dB{mB}1B{3B~1BzmoleB1BwB1tB0.025B1BB1@BB1dBmB1B3B1BmoleB1BvB1dBvB1BB1@BB1dBpB1pBB1 BB1dBvB1BB1tB400B1BKB1BNB3@Dxp9B1 pB1BB1dBW.34B1BB1+@B@FB1B _n_u_l_l_B3@D0B1 pB1 BB1dBQ.34B1BB1dB\DU.34B1BW.34B3@Dx/B1 pB1BB1dBQ.34B1BB1+@B@FB1B _n_u_l_l_B@DU%@MM5Process 4 to 1 is isochoric lowering of temperature. (5*5B-5B+"B- 5B-5B!@NormalTimes New Roman B3@DV56(B1 pB1 BB1dB\DU.41B1BB1dBNB1pBB1BB1@BB1dBuB1pBB1 BB1@BB1tB0.025B1BB1@BB1dBmB1B3B1BmoleB1BB1tB200B1BKB1BB1dBuB1pBB1 BB1@BB1tB0.025B1BB1@BB1dBmB1B3B1BmoleB1BB1tB400B1BKB3@Dp42(B1 pB1BB1dB\DU.41B1BB1+@B@FB1B _n_u_l_l_B3@DFcY3PB1 pB1 BB1dBW.41B1BB1tB0B1BjouleB3@Dx@YPB1 pB1BB1dBW.41B1BB1+@B@FB1B _n_u_l_l_B3@Dj0xB1 pB1 BB1dBQ.41B1BB1dB\DU.41B1BW.41B3@Dp>B1 pB1BB1dBQ.41B1BB1+@B@FB1B _n_u_l_l_B@D@@RWe now compute the total work done on the gas and the total heat added to the gas.(R*RB-RB+"B- RB-RB!@NormalTimes New Roman B3@D4B1 pB1 BB1dBW.totB1BB1@BC1@BC1dCW.12C1CW.23C1BW.34C1BW.41C3@D0SC1 pC1CC1dCW.totC 1CC 1+@C @FC 1C  _n_u_l_l_C 3@D16C 1 pC1 C C1dCQ.totC1CC1@CC1@CC1dCQ.12C1CQ.23C1CQ.34C1CQ.41C3@D0P7C1 pC1CC1dCQ.totC1CC1+@C@FC1C _n_u_l_l_C@D +Q  98+D+DARThe work on the gas is negative, i.e. the gas is doing positive work on the piston. The heat is positive, i.e. heat is added to the gas. Finally, we compute the efficiency of the heat engine: e = output/input. The output is the total work done on pistons: -Wtot. The input is the heat received during the isothermal expansion Q34.(Q RC- C )CC!-C")C_CC#-CC$)CC!C%-C&)C_C#C'-C()CC%C'C!*RC)-RC*+"C+- RC,-RC-!@NormalTimes New Roman C.3@Da U %x C/1 pC01 C/C11dC0eC21C0C31K@C2C41C3W.totC51C2Q.34C63@Dn } x C71 pC81C7C91dC8eC:1C8C;1+@C:@FC<1C: _n_u_l_l_C=@D H H  COPYRIGHT MIRON KAUFMAN 1997 (*C>-C?+"C@- CA-CB!@NormalTimes New Roman