CarnotVDWB.nb 1. Carnot Knowledge. Hal Harris Department of Chemistry University of Missouri-St. Louis St. Louis, Missouri

Transcription

1 CarnotVDWB.nb 1 Carnot Knowledge This is the second part of this worksheet and describes the Carnot cycle for methane Hal Harris Department of Chemistry University of Missouri-St. Louis St. Louis, Missouri hharris@umsl.edu Copyright 2001 by the Division of Chemical Education, Inc., American Chemical Society. All rights reserved. For classroom use by teachers, one copy per student in the class may be made free of charge. Write to JCE Online, jceonline@chem.wisc.edu, for permission to place a document, free of charge, on a class Intranet. Translated from Mathcad to Mathematica by: Laura Rachel Yindra, Journal of Chemical Education, University of Wisconsin-Madison, August Carnot cycle for methane In this second notebook, we'll develop the Carnot cycle for methane, as a van der Waals gas, in a parallel to what was done in Part 1 of "Carnot Knowledge." Students should have mastered the concepts in Part 1 and completed the exercises at the end of that notebook. Here in Part 2 we develop the method using an equation of state for real gases. After completing this notebook a student can create notebooks for other gases using other equations of state. The vdw equation of state is: P:=R*T / (V-b) - a / V^2 We will use the same variable names, except that the subscript labels will begin with VDW These are the vdw constants for methane 5 : In[1]:= << Miscellaneous`Units`

2 CarnotVDWB.nb 2 In[2]:= a:= Convert@2.283 Atmosphere Liter 2, Bar Liter 2 D In[3]:= b:= Liter 10 2 and its heat capacity 5 at 298 K is In[4]:= C PVDW := Joule Kelvin In[5]:= P1 := 20 Bar; P2 := 4 Bar In[6]:= R:= Liter Bar Kelvin Step a In the first, isothermal step of the cycle, we need the initial and final volumes of the gas, as the pressure changes from P1 to P2. The vdw equation is not as convenient for calculating volume as it ispressure, since there is the possibility of three different roots. We will write the equation of state in polynomial form, and find roots corresponding to P1 and P2. In[7]:= T hot := 400 Kelvin In[8]:= f@v_d := ab+ av HR T hot + P1 bl V 2 + P1 V 3 In[9]:= In[10]:= Out[10]= V VDW1 := x ê. Solve@f@xD 0, xd V VDW1 = V Liter In[11]:= f@v_d := ab+ av HR T hot + P2 bl V 2 + P2 V 3 In[12]:= In[13]:= Out[13]= In[14]:= V VDW2 := x ê. Solve@f@xD 0, xd V VDW2 = V Liter i:= Range@0, 100D In[15]:= v VDW12 := v VDW12 = V VDW i HV VDW2 V VDW1 L General::spell1 : Possible spelling error: new symbol name "VDW12" is similar to existing symbol "VDW2". More Convert and strip units to graph In[16]:= p VDW12 := p VDW12 = R T hot v VDW12 b a v2 VDW12

3 CarnotVDWB.nb 3 At each volume, we calculate the corresponding pressure. In[17]:= In[18]:= Out[18]= In[19]:= ta := ta = TableA9 v VDW12PxT 10 3 Liter, p VDW12PxT =, 8x, 1, 101<E Bar $TextStyle = FontColor RGBColor@0, 0, 0D FontColor RGBColor@0, 0, 0D << Graphics`FilledPlot` In[20]:= plot1 = FilledListPlot@ta, Fills RGBColor@1, 0, 0D, PlotRange 80, 2 10^6<, Frame True, FrameLabel 8"v VDW12 ", "p VDW12 "<D p VDW v VDW12 Out[20]= In[21]:= Out[21]= w VDW12 = ConvertA 2 Hp VDW12PxT + p VDW12 Px + 1TL Hv VDW12 Px + 1T v VDW12 PxTL, JouleE x= Joule Interlude: Derivations In this section of the notebook are derivations of equations that are needed for the calculations involving nonideal gases. In order to find the relationship between P and V in a reversible adiabatic change, we need an expression for dp/dv at constant S in terms of the measured properties of the fluid. See Reference (1), pp

4 CarnotVDWB.nb 4 HdP ê dvl S ê HdP ê dvl T ê dvl P HdP ê dsl V D ê dvl P HdP ê dtl V ] = HdS ê dtl P ê HdS ê dtl V =g where g = Cp/Cv Here we used the mathematical relationships: Hdz ê dxl y Hdx ê dyl z Hdy ê dzl x = -1 and Hdw ê dxl y / Hdz ê dxl y = Hdw ê dzl y ] Therefore, HdP ê dvl S = g HdP ê dvl T and this equation can be integrated step-by-step. For an ideal gas, PV = RT and HdP ê dvl S = -g P/V, whereas for a VdW gas, p = RT v b a v 2 dp dv = RT Hv bl 2 + 2a v 3 at constant T (isothermal) dp dv = RTγ Hv bl 2 at constant S (adiabatic) + 2aγ v 3 We also need HdT ê dvl S so that we can integrate both T and P versus volume. From Reference (2), p. 108, HdT ê dpl S = - HdS ê dpl T / HdS ê dtl P = T/Cp HdV ê dtl P and HdT ê dvl S = - HdS ê dvl T / HdS ê dtl V = -T/Cv HdP ê dtl V. For a VdW gas, HdP ê dtl V = R / (V-b), so that HdT ê dvl S = - RT/Cv(V-b). dt dv = RT Cv Hv bl T cool1 V3 ê T T = RêCv 1 êhv bl v T hot V2 LogA T V3 b cool R Log@ E = V2 b D Cv T hot T cool = V3 b R LogA V2 b E Cv T hot This is the equation used to calculate the intermediate temperatures (T cool ) during the adiabatic steps for the vdw gas.

5 CarnotVDWB.nb 5 End Interlude Step b The exponent "3/2" in the equation for V3 is the ratio Cv/R for a monatomic gas. for methane, Cp=35.31 J/K mol at 298K; Cv=Cd-R (This is apporzimately true for real gases as well as ideal ones. See Reference 2, p.107.) In[22]:= In[23]:= Cv := Cv = Convert@C PVDW, Bar Liter ê KelvinD R Cv ê R Convert::temp : Warning: Convert@old,newD converts units of temperature. ConvertTemperature@temp,old,newD converts absolute temperature. Out[23]= The heat capacities are actually temerature-dependent, but we'll neblect that. (It would not be too difficult to incorporate.) Cv / R Log[T cole /T hot ]=Log[V2 / V3] In[24]:= In[25]:= Out[25]= T cold := 300 Kelvin V VDW3 = V VDW2 J T CvêR hot N T cold Liter In[26]:= v VDW23 := v VDW23 = V VDW2 + i V VDW3 V VDW2 100 General::spell1 : Possible spelling error: new symbol name "VDW23" is similar to existing symbol "VDW3". More As the volume changes from V VDW2 to V VDW3, the temperature and the pressure both change. The relationships needed are derived in the column at the right. In[27]:= T VDW23 := T VDW23 = R LogA v VDW23 b V VDW2 b E Cv T hot In[28]:= p VDW23 := p VDW23 = R T VDW23 v VDW23 b a v2 VDW23 In[29]:= Out[29]= plot2 = FilledListPlot@ Table@8v 10^ 3 Liter^ 1, p Bar^ 1<, 8x, 1, 101<D, Fills RGBColor@0, 0, 1D, DisplayFunction IdentityD

6 CarnotVDWB.nb 6 In[30]:= Show@plot1, plot2, DisplayFunction $DisplayFunction, FrameLabel 8"v VDW12,v VDW23 ", "p VDW12,p VDW23 "<D p V DW1 2,p VDW v VDW12,v VDW23 Out[30]= In[31]:= Out[31]= w VDW23 = ConvertA Joule j=1 2 Hp VDW23PjT + p VDW23 Pj + 1TL Hv VDW23 Pj + 1T v VDW23 PjTL, JouleE Step c In[32]:= Out[32]= V VDW4 = V VDW1 J T CvêR hot N T cold Liter In[33]:= Out[33]= In[34]:= V VDW Liter j:= Range@0, 200D In[35]:= v VDW34 := v VDW34 = V VDW j HV VDW3 V VDW4 L General::spell1 : Possible spelling error: new symbol name "VDW34" is similar to existing symbol "VDW4". More In[36]:= p VDW34 := p VDW34 = R T cold v VDW34 b a v2 VDW34 In[37]:= Out[37]= w VDW34 = ConvertA 2 Hp VDW34PxT + p VDW34 Px + 1TL Hv VDW34 Px + 1T v VDW34 PxTL, JouleE x= Joule

7 CarnotVDWB.nb 7 In[38]:= Out[38]= In[39]:= plot3 = FilledListPlot@ Reverse@Table@8v 10^ 3 Liter^ 1, p Bar^ 1<, 8x, 1, 201<DD, Fills RGBColor@1, 1, 1D, DisplayFunction IdentityD Show@plot1, plot2, plot3, DisplayFunction $DisplayFunction, FrameLabel 8"v VDW12,v VDW23,v VDW34 ", "p VDW12,p VDW23,p VDW34 "<D p V DW1 2,p V DW2 3,p VDW v VDW12,v VDW23,v VDW34 Out[39]= Step d In[40]:= In[41]:= Meter3 V1 := ConvertA, LiterE 10 3 v VDW41 := v VDW41 = V VDW4 i V VDW4 V1 99 General::spell1 : Possible spelling error: new symbol name "VDW41" is similar to existing symbol "VDW1". More In[42]:= T VDW41 := T VDW41 = R LogA v VDW41 b V VDW4 b E Cv T cold In[43]:= p VDW41 := p VDW41 = R T VDW41 v VDW41 b a v2 VDW41 In[44]:= Out[44]= plot4 = FilledListPlot@ Reverse@Table@8v 10^ 3 Liter^ 1, p Bar^ 1<, 8x, 1, 101<DD, Fills RGBColor@1, 1, 1D, DisplayFunction IdentityD

8 CarnotVDWB.nb 8 In[45]:= Show@plot1, plot2, plot3, plot4, DisplayFunction $DisplayFunction, FrameLabel 8"v VDW12,v VDW23,v VDW34,v VDW41 ", "p VDW12,p VDW23,p VDW34,p VDW41 "<D p V DW1 2,p V DW2 3,p V DW3 4,p VDW v VDW12,v VDW23,v VDW34,v VDW41 Out[45]= In[46]:= In[47]:= Out[47]= 99 1 w VDW41 := w VDW41 = ConvertA 2 Hp VDW41PxT + p VDW41 Px + 1TL Hv VDW41 Px + 1T v VDW41 PxTL, JouleE x=1 w VDWnet = w VDW12 + w VDW23 + w VDW34 + w VDW Joule In[48]:= Out[48]= w VDW Joule In[49]:= Out[49]= w VDW Joule In[50]:= Out[50]= w VDW Joule In[51]:= Out[51]= w VDW Joule Because q = -w for step a, we could use -w VDW12. The calculation at left is an alternative evaluation. In[52]:= Out[52]= V VDW2J RT hot q VDWh = ConvertA V VDW1 v b N v, JouleE i V VDW2 j k V VDW1 Liter^ Joule Liter^ 1 i a Bar 1 Liter 2 y j k HvL 2 z vy z 100 Joule { { In[53]:= Out[53]= w VDW Joule

9 CarnotVDWB.nb 9 In[54]:= ξ VDW = Abs@w VDWnet Joule^ 1D Joule q VDWh Out[54]= The difference in efficiency between this and the monatomic ideal gas is presumably a reflection of numerical integration error, the approximations that the heat capacity is constant, and that Cp-Cv=R. Of course, both of them should be given by e=1-t cold /T hot =1-300/400=0.25 One of the interesting aspects of the comparison between ideal and vdw gases is that, while the efficiency is determined by the temperatures and is equal for the two cases, both of the the adiabatic work terms are larger for the vdw gas. Which parameter of the model causes this to be so? References 1. A. B. Pippard, The Elements of Classical Thermodynamics (Cambridge University Press, 1961) 2. Gilbert Newton Lewis and Merle Randall (revised by Kenneth S. Pitzer and Leo Brewer) Thermodynamics, Second Edition (Mc Graw-Hill, 1961) 3. Robert G. Mortimer Physical Chemistry, Second Edition (Harcourt Academic Press, 2000) 4. R. Stephen Berry, Stuart A. Rice, and John Ross Physical Chemistry, Second Edition, (Oxford University Press, 2000) 5. Peter Atkins Physical Chemistry, Sixth Edition (W. H. Freeman, 1998) 6. Walter Dannhauser, "PVT Behavior of Real Gases" Journal of Chemical Education J. L. Pauley and Elwyn H. Davis, "P-V-T Isotherms of Real Gases", Journal of Chemical Education In[55]:= ClearAll@R, P1, a, b, f, i, ta, plot1, Cv, plot2, j, plot3, V1, plot4d