DOCSLIB.ORG

Math Background for Thermodynamics ∑

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Math Background for Thermodynamics ∑ MATH BACKGROUND FOR THERMODYNAMICS A. Partial Derivatives and Total Differentials Partial Derivatives Given a function f(x1,x2,...,xm) of m independent variables, the partial derivative ∂ f of f with respect to x , holding the other m-1 independent variables constant, , is defined by i ∂ xi xj≠i ∂ f fx( , x ,..., x+ ∆ x ,..., x )− fx ( , x ,..., x ,..., x ) = 12ii m 12 i m ∂ lim ∆ xi x →∆ 0 xi xj≠i i nRT Example: If p(n,V,T) = , V ∂ p RT ∂ p nRT ∂ p nR = = − = ∂ n V ∂V 2 ∂T V VT,, nTV nV , Total Differentials Given a function f(x1,x2,...,xm) of m independent variables, the total differential of f, df, is defined by m ∂ f df = ∑ dx ∂ i i=1 xi xji≠ ∂ f ∂ f ∂ f = dx + dx + ... + dx , ∂ 1 ∂ 2 ∂ m x1 x2 xm xx2131,...,mm xxx , ,..., xx ,..., m-1 where dxi is an infinitesimally small but arbitrary change in the variable xi. nRT Example: For p(n,V,T) = , V ∂ p ∂ p ∂ p dp = dn + dV + dT ∂ n ∂ V ∂ T VT,,, nT nV RT nRT nR = dn − dV + dT V V 2 V B. Some Useful Properties of Partial Derivatives 1. The order of differentiation in mixed second derivatives is immaterial; e.g., for a function f(x,y), ∂ ∂ f ∂ ∂ f ∂ 22f ∂ f = or = ∂ y ∂ xx ∂ ∂ y ∂∂yx ∂∂xy y x x y 2 in the commonly used short-hand notation. (This relation can be shown to follow from the definition of partial derivatives.) 2. Given a function f(x,y): ∂ y 1 a. = etc. ∂ f ∂ f x ∂ y x ∂ f ∂ y ∂ x b. = −1 (the cyclic rule) ∂ x ∂ f ∂ y yxf Proof: The total differential of f is ∂ f ∂ f df = dx + dy ∂ x ∂ y yx We could, in principle, solve the equation f = f(x,y) for y in terms of x and f; i.e., we could obtain the function y(x,f). Its total differential would be given by ∂ y ∂ y dy = dx + df ∂ x ∂ f fx Substituting the above expression for df into the right-hand side of the expression for dy, we obtain ∂ y ∂ y ∂ f ∂ f dy = dx + dx + dy ∂ x ∂ f ∂ x ∂ y fxyx or ∂ y ∂ f ∂ y ∂ y ∂ f 0 = + dx + - 1dy (I) ∂ f ∂ x ∂ x ∂ f ∂ y xy f xx Since dx and dy are arbitrary infinitesimal changes, the only way that the right-handside of (I) can be zero for all choices of dx and dy is if the coefficients of dx and dy are each separately equal to zero. Setting the coefficient of dy equal to zero gives ∂ y ∂ f ∂ y 1 = 1 or = q.e.d. ∂ f ∂ y ∂ f ∂ f xx x ∂ y x 3 Moreover, since any of the three variables f, x, and y could have been chosen as the dependent variable, it must also be true that ∂ x 1 ∂ x 1 = and = ∂ f ∂ f ∂ y ∂ y y f ∂ x ∂ x y f Setting the coefficient of dx equal to zero in eqn. (I) gives ∂ y ∂ f ∂ y = - ∂ f ∂ x ∂ x xy f Thus, ∂ y ∂ f ∂ f ∂ x xy 1 ∂ x = -1 or since = , ∂ y ∂ y ∂ y f ∂ x ∂ x ff ∂ f ∂ y ∂ x − = 1 q.e.d. ∂ x ∂ f ∂ y yxf 1 ∂ V 1 ∂ V Example: Using the cyclic rule and the definitions α = and β = - , V ∂ T V ∂ p p T ∂ p α show that = . ∂T β V Solution: From the cyclic rule, ∂ p ∂ V ∂T = −1. ∂T ∂ p ∂ V VTp Thus, ∂ V - ∂ T ∂ p -1 p -Vα α = = = = q.e.d. ∂T ∂ V ∂T ∂ V -Vβ β V ∂ p ∂ V ∂ p Tp T 3. Given two functions of x and y: f(x,y) and g(x,y), ∂ f ∂ f ∂ y a. = (chain rule) ∂ g ∂ y ∂ g xxx 4 ∂ f ∂ f ∂ f ∂ y b. = + ∂ x ∂ x ∂ y ∂ x gxgy Proof: The total differential of f considered to be a function of x and y is ∂ f ∂ f df = dx + dy , ∂ x ∂ y yx while the total differential of y considered to be a function of x and g is ∂ y ∂ y dy = dx + dg . ∂ x ∂ g gx Substituting this expression for dy into the expression for df above gives ∂ f ∂ f ∂ y ∂ y df = dx + dx + dg ∂ x ∂ y ∂ x ∂ g yxgx ∂ f ∂ f ∂ y ∂ f ∂ y = + dx + dg. ∂ x ∂ y ∂ x ∂ y ∂ g yxg xx But if f is considered to be a function of x and g, rather than of x and y, its total differential is given by ∂ f ∂ f df = dx + dg. ∂ x ∂ g gx These two expressions for df must be identical for any choice of the arbitrary infinitesimal changes dx and dg. Therefore, it must be true that ∂ f ∂ f ∂ f ∂ y ∂ f ∂ f ∂ y = + and = q.e.d. ∂ x ∂ x ∂ y ∂ x ∂ g ∂ y ∂ g gxgy xxx Examples: 1. For a van der Waals gas (i.e., a model system which obeys the van der Waals equation of state), the pressure p and the internal energy U, as functions of n, V, and T, are given by 2 2 nRT na 3 na p = − and U = nRT − Vnb− V 2 2 V respectively, where a and b are constants. Use these equations and the chain rule to derive an ∂ U equation for in terms of n, V, and T. ∂ p nT, 5 ∂ U ∂ V ∂ U ∂ U ∂ V nT, Solution: = = ∂ p ∂ V ∂ p ∂ p nT,,, nT nT ∂ V nT, naV22/ na = = 22na2 nRT na RTV 2 − − V 32()Vnb− V ()Vnb− 2 ∂ U 2. Given that the constant-volume heat capacity C ≡ , show that V ∂ T V ∂U ∂U = CV + α . ∂T V ∂V p T ∂ U ∂ U ∂ U ∂ V Solution: = + ∂T ∂T ∂ V ∂ T pVTp ∂ U ∂ V But = C and = Vα. Therefore, ∂ T V ∂T V p ∂ U ∂ U = CV + α q.e.d. ∂ T V ∂ V p T ------------------------------------------------------------------------------------------------------- Note: In this section, we have considered functions of two independent variables. However, all the relations derived hold for functions of more than two variables provided all the additional independent variables are held constant; e.g., for a function f(w,x,y,z), ∂ ∂ f ∂ ∂ f = , ∂x ∂ yy ∂ ∂ x wxz,, wyz,, wyz,, wxz,, ∂ f ∂ y ∂ x = − 1, etc. ∂ x ∂ f ∂ y wyz,, wxz ,, f ,, wz 6 C. Exact and Inexact Differentials; Line Integrals Linear Differentials An infinitesimal quantity m dz = ∑ Mi (, x12 x ,...,) xmi dx i=1 = M11 dx + M 2 dx 2 + ... + Mmm dx is called a linear differential; the right-hand side of the equation is called a linear differential form in m variables. We shall be concerned primarily with linear differentials in two variables; in this case, let us write dz = Mdx + Ndy where M and N are, in general, functions of x and y. Exact Differentials: dz = Mdx + Ndy is an exact differential if and only if there exists a function of x and y, F(x,y), such that dF = dz for all values of x and y. A more practically useful test for exactness is the following: dz = Mdx + Ndy is an exact differential ∂ M ∂ N if and only if = . ∂ y ∂ x xy It is easy to see that this relation must be true for an exact differential dz because in that case ∂ F ∂ F M = and N = and ∂ x ∂ y y x ∂ M ∂ ∂ F ∂ ∂ F ∂ N = = = . ∂ yy ∂ ∂ xx ∂ ∂ y ∂ x xy x y x y ∂ M ∂ N It can also be shown that the converse is true; i.e., if = , then dz is exact. ∂ y ∂ x xy Inexact Differentials dw = M′dx + N′dy is an inexact differential if and only if there exists no function F(x,y), such that dF = dw for all values of x and y. Equivalently, dw is inexact if and only if ∂ M '' ∂ N ≠ . ∂ y ∂ x xy Examples: The linear differential dz = xy2dx + x2ydy is an exact differential because it is the total differential of the function F = ½ x2y2 + C, where C is an arbitrary constant. Here, ∂ M ∂ ∂ ∂ N = ()xy22 = 2. xy = ()xy = ∂ yy ∂ ∂ x ∂ x xx yy 7 2 2 On the other hand, the linear differential dw = x y dx + xy dy is inexact, as can most easily be demonstrated by noting that ∂ M ∂ ∂ N ∂ = ()xy222 = x2 while = ()xy = y ∂ yy ∂ ∂ xx ∂ xx y y Line Integrals Consider a curve C in the X-Y plane connecting two points (x′,y′) and (x′′,y′′) and imagine that we chop the curve up into n segments as shown at the left. Then the line integral I of the linear differential dz = Mdx + Ndy on the curve C is defined by n []∆∆+ I = lim∑ MxyxNxyy(,ii ) i (, ii ) i n→∞ i=1 where M(x,y) and N(x,y) are given functions and M(xi,yi) and N(xi, yi) are the values of these functions at the midpoint of the ith segment. The notation we shall use for this integral is ('','')xy I =[∫ Mdx+ Ndy] (',')xy C In order to carry out this integration, one must eliminate one of the variables x or y, which are related via the equation for the curve C, and then evaluate the resulting ordinary definite integral. Examples: 1. Calculate the line integral of the inexact differential dw = x2ydx + xydy along the curve y = x from (0,0) to (1,1). (,)11 Solution: I =[∫ x2 dx+ xydy] But y = x and dy = dx. Therefore, (,)00 yx= 1 1 2 2 3 2 I = 2∫ x dx = 3 x = 3 0 0 2. Calculate the line integral of dw = x2ydx + xydy along the curve y = x2 from (0,0) to (1,1). (,)11 Solution: I =[∫ x2 dx+ xydy] But y = x2 and dy = 2xdx.
Load more

Recommended publications
  • Thermodynamic Potentials and Natural Variables
    Revista Brasileira de Ensino de Física, vol. 42, e20190127 (2020) Articles www.scielo.br/rbef cb DOI: http://dx.doi.org/10.1590/1806-9126-RBEF-2019-0127 Licença Creative Commons Thermodynamic Potentials and Natural Variables M. Amaku1,2, F. A. B. Coutinho*1, L. N. Oliveira3 1Universidade de São Paulo, Faculdade de Medicina, São Paulo, SP, Brasil 2Universidade de São Paulo, Faculdade de Medicina Veterinária e Zootecnia, São Paulo, SP, Brasil 3Universidade de São Paulo, Instituto de Física de São Carlos, São Carlos, SP, Brasil Received on May 30, 2019. Revised on September 13, 2018. Accepted on October 4, 2019. Most books on Thermodynamics explain what thermodynamic potentials are and how conveniently they describe the properties of physical systems. Certain books add that, to be useful, the thermodynamic potentials must be expressed in their “natural variables”. Here we show that, given a set of physical variables, an appropriate thermodynamic potential can always be defined, which contains all the thermodynamic information about the system. We adopt various perspectives to discuss this point, which to the best of our knowledge has not been clearly presented in the literature. Keywords: Thermodynamic Potentials, natural variables, Legendre transforms. 1. Introduction same statement cannot be applied to the temperature. In real fluids, even in simple ones, the proportionality Basic concepts are most easily understood when we dis- to T is washed out, and the Internal Energy is more cuss simple systems. Consider an ideal gas in a cylinder. conveniently expressed as a function of the entropy and The cylinder is closed, its walls are conducting, and a volume: U = U(S, V ).
    [Show full text]
  • 6CCP3212 Statistical Mechanics Homework 1
    6CCP3212 Statistical Mechanics Homework 1 Lecturer: Dr. Eugene A. Lim 2018-19 Year 3 Semester 1 https://nms.kcl.ac.uk/eugene.lim/teach/statmech/sm.html 1) (i) For the following differentials with α and β non-zero real constants, which are exact and which are inexact? Integrate the equation if it is exact. (a) x dG = αdx + β dy (1) y (b) α dG = dx + βdy (2) x (c) x2 dG = (x + y)dx + dy (3) 2 (ii) Show that the work done on the system at pressure P d¯W = −P dV (4) where dV is the change in volume is an inexact differential by showing that there exists no possible function of state for W (P; V ). (iii) Consider the differential dF = (x2 − y)dx + xdy : (5) (a) Show that this is not an exact differential. And hence integrate this equation in two different straight paths from (1; 1) ! (2; 2) and from (1; 1) ! (1; 2) ! (2; 2), where (x; y) indicates the locations. Compare the results { are they identical? (b) Define a new differential with dF y 1 dG ≡ = 1 − dx + dy : (6) x2 x2 x Show that dG is exact, and find G(x; y). 2) This problem asks you to derive some derivative identities of a system with three variables x, y and z, with a single constraint x(y; z). This kind of system is central to thermodynamics as we often use three state variables P , V and T , with an equation of state P (V; T ) (i.e. the constraint) to describe a system.
    [Show full text]
  • 3 More Applications of Derivatives
    3 More applications of derivatives 3.1 Exact & inexact di®erentials in thermodynamics So far we have been discussing total or \exact" di®erentials µ ¶ µ ¶ @u @u du = dx + dy; (1) @x y @y x but we could imagine a more general situation du = M(x; y)dx + N(x; y)dy: (2) ¡ ¢ ³ ´ If the di®erential is exact, M = @u and N = @u . By the identity of mixed @x y @y x partial derivatives, we have µ ¶ µ ¶ µ ¶ @M @2u @N = = (3) @y x @x@y @x y Ex: Ideal gas pV = RT (for 1 mole), take V = V (T; p), so µ ¶ µ ¶ @V @V R RT dV = dT + dp = dT ¡ 2 dp (4) @T p @p T p p Now the work done in changing the volume of a gas is RT dW = pdV = RdT ¡ dp: (5) p Let's calculate the total change in volume and work done in changing the system between two points A and C in p; T space, along paths AC or ABC. 1. Path AC: dT T ¡ T ¢T ¢T = 2 1 ´ so dT = dp (6) dp p2 ¡ p1 ¢p ¢p T ¡ T1 ¢T ¢T & = ) T ¡ T1 = (p ¡ p1) (7) p ¡ p1 ¢p ¢p so (8) R ¢T R ¢T R ¢T dV = dp ¡ [T + (p ¡ p )]dp = ¡ (T ¡ p )dp (9) p ¢p p2 1 ¢p 1 p2 1 ¢p 1 R ¢T dW = ¡ (T ¡ p )dp (10) p 1 ¢p 1 1 T (p ,T ) 2 2 C (p,T) (p1,T1) A B p Figure 1: Path in p; T plane for thermodynamic process.
    [Show full text]
  • Thermodynamics
    ME346A Introduction to Statistical Mechanics { Wei Cai { Stanford University { Win 2011 Handout 6. Thermodynamics January 26, 2011 Contents 1 Laws of thermodynamics 2 1.1 The zeroth law . .3 1.2 The first law . .4 1.3 The second law . .5 1.3.1 Efficiency of Carnot engine . .5 1.3.2 Alternative statements of the second law . .7 1.4 The third law . .8 2 Mathematics of thermodynamics 9 2.1 Equation of state . .9 2.2 Gibbs-Duhem relation . 11 2.2.1 Homogeneous function . 11 2.2.2 Virial theorem / Euler theorem . 12 2.3 Maxwell relations . 13 2.4 Legendre transform . 15 2.5 Thermodynamic potentials . 16 3 Worked examples 21 3.1 Thermodynamic potentials and Maxwell's relation . 21 3.2 Properties of ideal gas . 24 3.3 Gas expansion . 28 4 Irreversible processes 32 4.1 Entropy and irreversibility . 32 4.2 Variational statement of second law . 32 1 In the 1st lecture, we will discuss the concepts of thermodynamics, namely its 4 laws. The most important concepts are the second law and the notion of Entropy. (reading assignment: Reif x 3.10, 3.11) In the 2nd lecture, We will discuss the mathematics of thermodynamics, i.e. the machinery to make quantitative predictions. We will deal with partial derivatives and Legendre transforms. (reading assignment: Reif x 4.1-4.7, 5.1-5.12) 1 Laws of thermodynamics Thermodynamics is a branch of science connected with the nature of heat and its conver- sion to mechanical, electrical and chemical energy. (The Webster pocket dictionary defines, Thermodynamics: physics of heat.) Historically, it grew out of efforts to construct more efficient heat engines | devices for ex- tracting useful work from expanding hot gases (http://www.answers.com/thermodynamics).
    [Show full text]
  • Vector Calculus and Differential Forms with Applications To
    Vector Calculus and Differential Forms with Applications to Electromagnetism Sean Roberson May 7, 2015 PREFACE This paper is written as a final project for a course in vector analysis, taught at Texas A&M University - San Antonio in the spring of 2015 as an independent study course. Students in mathematics, physics, engineering, and the sciences usually go through a sequence of three calculus courses before go- ing on to differential equations, real analysis, and linear algebra. In the third course, traditionally reserved for multivariable calculus, stu- dents usually learn how to differentiate functions of several variable and integrate over general domains in space. Very rarely, as was my case, will professors have time to cover the important integral theo- rems using vector functions: Green’s Theorem, Stokes’ Theorem, etc. In some universities, such as UCSD and Cornell, honors students are able to take an accelerated calculus sequence using the text Vector Cal- culus, Linear Algebra, and Differential Forms by John Hamal Hubbard and Barbara Burke Hubbard. Here, students learn multivariable cal- culus using linear algebra and real analysis, and then they generalize familiar integral theorems using the language of differential forms. This paper was written over the course of one semester, where the majority of the book was covered. Some details, such as orientation of manifolds, topology, and the foundation of the integral were skipped to save length. The paper should still be readable by a student with at least three semesters of calculus, one course in linear algebra, and one course in real analysis - all at the undergraduate level.
    [Show full text]
  • Notes on the Calculus of Thermodynamics
    Supplementary Notes for Chapter 5 The Calculus of Thermodynamics Objectives of Chapter 5 1. to understand the framework of the Fundamental Equation – including the geometric and mathematical relationships among derived properties (U, S, H, A, and G) 2. to describe methods of derivative manipulation that are useful for computing changes in derived property values using measurable, experimentally accessible properties like T, P, V, Ni, xi, and ρ . 3. to introduce the use of Legendre Transformations as a way of alternating the Fundamental Equation without losing information content Starting with the combined 1st and 2nd Laws and Euler’s theorem we can generate the Fundamental Equation: Recall for the combined 1st and 2nd Laws: • Reversible, quasi-static • Only PdV work • Simple, open system (no KE, PE effects) • For an n component system n dU = Td S − PdV + ∑()H − TS i dNi i=1 n dU = Td S − PdV + ∑µidNi i=1 and Euler’s Theorem: • Applies to all smoothly-varying homogeneous functions f, f(a,b,…, x,y, … ) where a,b, … intensive variables are homogenous to zero order in mass and x,y, extensive variables are homogeneous to the 1st degree in mass or moles (N). • df is an exact differential (not path dependent) and can be integrated directly if Y = ky and X = kx then Modified: 11/19/03 1 f(a,b, …, X,Y, …) = k f(a,b, …, x,y, …) and ⎛ ∂f ⎞ ⎛ ∂f ⎞ x⎜ ⎟ + y⎜ ⎟ + ... = ()1 f (a,b,...x, y,...) ⎝ ∂x ⎠a,b,...,y,.. ⎝ ∂y ⎠a,b,..,x,.. Fundamental Equation: • Can be obtained via Euler integration of combined 1st and 2nd Laws • Expressed in Energy (U) or Entropy (S) representation n U = fu []S,V , N1, N2 ,..., Nn = T S − PV + ∑µi Ni i=1 or n U P µi S = f s []U,V , N1, N2 ,..., Nn = + V − ∑ Ni T T i=1 T The following section summarizes a number of useful techniques for manipulating thermodynamic derivative relationships Consider a general function of n + 2 variables F ( x, y,z32,...,zn+ ) where x ≡ z1, y ≡ z2.
    [Show full text]
  • Physical Chemistry II “The Mistress of the World and Her Shadow” Chemistry 402
    Physical Chemistry II “The mistress of the world and her shadow” Chemistry 402 L. G. Sobotka Department of Chemistry Washington University, St Louis, MO, 63130 January 3, 2012 Contents IIntroduction 7 1 Physical Chemistry II - 402 -Thermodynamics (mostly) 8 1.1Who,when,where.............................................. 8 1.2CourseContent/Logistics.......................................... 8 1.3Grading.................................................... 8 1.3.1 Exams................................................. 8 1.3.2 Quizzes................................................ 8 1.3.3 ProblemSets............................................. 8 1.3.4 Grading................................................ 8 2Constants 9 3 The Structure of Physical Science 10 3.1ClassicalMechanics.............................................. 10 3.2QuantumMechanics............................................. 11 3.3StatisticalMechanics............................................. 11 3.4Thermodynamics............................................... 12 3.5Kinetics.................................................... 13 4RequisiteMath 15 4.1 Exact differentials.............................................. 15 4.2Euler’sReciprocityrelation......................................... 15 4.2.1 Example................................................ 16 4.3Euler’sCyclicrelation............................................ 16 4.3.1 Example................................................ 16 4.4Integratingfactors.............................................. 17 4.5LegendreTransformations.........................................
    [Show full text]
  • Exact and Inexact Differentials in the Early Development of Mechanics
    Revista Brasileira de Ensino de Física, vol. 42, e20190192 (2020) Articles www.scielo.br/rbef cb DOI: http://dx.doi.org/10.1590/1806-9126-RBEF-2019-0192 Licença Creative Commons Exact and inexact differentials in the early development of mechanics and thermodynamics Mário J. de Oliveira*1 1Universidade de São Paulo, Instituto de Física, São Paulo, SP, Brasil Received on July 31, 2019. Revised on October 8, 2019. Accepted on October 13, 2019. We give an account and a critical analysis of the use of exact and inexact differentials in the early development of mechanics and thermodynamics, and the emergence of differential calculus and how it was applied to solve some mechanical problems, such as those related to the cycloidal pendulum. The Lagrange equations of motions are presented in the form they were originally obtained in terms of differentials from the principle of virtual work. The derivation of the conservation of energy in differential form as obtained originally by Clausius from the equivalence of heat and work is also examined. Keywords: differential, differential calculus, analytical mechanics, thermodynamics. 1. Introduction variable x. If another variable y depends on the indepen- dent variable x, then the resulting increment dy of y is It is usual to formulate the basic equations of thermo- its differential. The quotient of these two differentials, dynamics in terms of differentials. The conservation of dy/dx, was interpreted geometrically by Leibniz as the energy is written as ratio of the ordinate y of a point on a curve and the length of the subtangent associated to this point.
    [Show full text]
  • W. M. White Geochemistry Appendix II Summary of Important Equations
    W. M. White Geochemistry Appendix II Summary of Important Equations Equations of State: Ideal GasLaw: PV = NRT Coefficient of Thermal Expansion: ∂ α≡α ≡ 1 V V ∂T Compressibility: ∂V β≡β≡± 1 V ∂P Van der Waals Equation: RT a P= - 2 V ± b V The Laws of Thermdynamics: First Law: ∆U = Q + W 1 written in differential form: dU = dQ + dW 2 work done on the system and heat added to the system are positive. The first law states the equivalence of heat and work and the conservation of energy. Second Law: dQrev = TdS 3 Two ways of stating the second law are Every system left to itself will, on average, change to a condition of maximum probability and Heat cannot be extracted from a body and turned entirely into work. Third Law: lim S= 0 T → 0 3 This follows from the facts that S = R ln Ω and Ω =1 at T = 0 for a perfectly crystalline pure substance. Primary Variables of Thermodynamics The leading thermodynamic properties of a fluid are determined by the relations which exist between the volume, pressure, termperature, energy and entropy of a given mass of fluid in a state of thermodynamic equilibrium - J. W. Gibbs The primary variables of thermodynamics are P, V, T, U, and S. Other thermodynamic functions can be stated in terms of these variables. For various combination of these variables there are 1 W. M. White Geochemistry Appendix II Equation Summary characteristics functions. The characteristic function for S and V is one of the primary variables: U. Thus dU = TdS + PdV 5 Other Important Thermodynamic Functions What then is the use of thermodynamic equations? They are useful precisely because some quantities are easier to measure than others.
    [Show full text]
  • Basic Thermodynamics
    Basic Thermodynamics Handout 1 First Law of Thermodynamics Basically a statement of conservation of energy which includes heat as a form of energy: dU = ¯dQ + ¯dW; where ¯dQ is heat absorbed by the system, and ¯dW is work done on the system. U is a function of state called internal energy. Zeroth Law of Thermodynamics Suppose A, B and C are distinct thermodynamic systems. If A is in thermal equilibrium with B, and B is in thermal equilibrium with C, then A is in thermal equilibrium with C. Systems in thermodynamic equilibrium can be described in terms of physical variables that do not change with time. The Zeroth Law implies that there exists some physical property which systems that are in thermal equilibrium have in common, regardless of their size, shape, substance, etc. We call this property temperature.A thermometer is a device that can be used to assign a numerical value to temperature through measurements of a physical property. Constant-volume gas thermometer. The mercury reservoir is raised or lowered to adjust the difference in height h between the two mer- cury columns M and M 0 so that the volume of gas in the bulb + capillary is constant. Absolute temperature (perfect gas scale) lim ! (pV ) T (K) = 273:16 p 0 T ; limp!0(pV )TP where TP stands for the triple point of water. Right: Readings of a constant- volume gas thermometer for the temperature of condensing steam as a function of the pressure p3 at the triple point of water. Curves for different gases are shown.
    [Show full text]
  • The First Law: the Machinery the Power of Thermodynamics – How to Establish Relations Between Different Properties of a System
    The First Law: the machinery The power of thermodynamics – how to establish relations between different properties of a system. The procedure is based on the experimental fact that the internal energy and the enthalpy are state functions. We shall see that a property can be measured indirectly by measuring others and then combining their values. We shall also discuss the liquefaction of gases and establish the relation between Cp and CV. State functions and exact differentials State and path functions The initial state of the system is i and in this state the internal energy is Ui. Work is done by the system as it expands adiabatically to a state f (an internal energy Uf). The work done on the system as it changes along Path 1 from i to f is w. U – a property of the state; w – a property of the path. Consider another process, Path 2: the initial and final states are the same, but the expansion is not adiabatic. Because U is a state function, the internal energy of both the initial and the final states are the same as before, but an energy q’ enters the system as heat and the work w’ is not the same as w. The work and the heat are path functions. Exact and inexact differentials If a system is taken along a path, U changes from Ui to Uf and the overall change is the sum (integral) of all infinitesimal changes along the path: f ΔU = dU ∫i ΔU depends on the initial and final states but is independent of the path between them.
    [Show full text]
  • Seoul National University Prof. Sang-Im, Yoo 2.7 Standard Enthalpy Changes the Enthalpy Change for a Process in the Standard States
    Lecture Contents 2.2. TheThe FirstFirst LawLaw Thermochemistry 2.7 Standard enthalpy changes 2.8 Standard enthalpies of formation 2.9 The temperature-dependence of reaction enthalpies State functions and exact differentials 2.10 Exact and inexact differentials 2.11 Changes in internal energy 2.12 The Joule-Thomson effect Seoul National University Prof. Sang-im, Yoo 2.7 Standard enthalpy changes The enthalpy change for a process in the standard states ► The standard state of a substance: Its pure form at a specific temperature and 1 bar ex) The standard state of ethanol at 298 K : a pure liquid ethanol at 298 K and 1 bar The standard state of iron at 500 K : a pure solid iron at 500 K and 1 bar ► Standard enthalpy change for a reaction: The difference between the products in their standard states and the reactants in their standard states at the same specified temperature o ex) The standard enthalpy of vaporization, ΔH vap The enthalpy change per mole when a pure liquid at 1 bar vaporizes to gas at 1 bar o H2O(l) → H2O(g) ΔH vap(373 K) = +40.66 kJ/mol - Conventional temperature for reporting thermodynamic data? 298.15K (25.00oC) Contents (a) Enthalpies of physical change (b) Enthalpies of chemical change (c) Hess’s law Seoul National University Prof. Sang-im, Yoo 2.7 Standard enthalpy changes (a) Enthalpies of physical change o - Standard enthalpy of transition, ΔH trs: the standard enthalpy change that accompanies change of physical state (see Table 2.3) o ex) Standard enthalpy of vaporization, ΔH vap o Standard enthalpy of fusion, ΔH fus o H2O(l) → H2O(s) ΔH fus(273 K) = +6.01 kJ/mol - Enthalpy is a state function (Independent of the path between the two states) (i) Directly from solid to vapor: sublimation o H2O(s) → H2O(g) ΔH sub (ii) In two steps : fusion + vaporization o H2O(s) → H2O(l) ΔH fus o H2O(l) → H2O(g) ΔH vap o o o H2O(s) → H2O(g) ΔH sub = ΔH fus + ΔH vap - The standard enthalpy changes of a forward process and its reverse differ only in sign ΔHo(A → B) = − ΔHo(B → A) Seoul National University Prof.
    [Show full text]