06 - G eneral Thermodynamic Relationships single component systems o r systems of constant com position The relationships which follow are based on

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General Thermodynamic Relationships: single component systems, or systems of constant com position The relationships which follow are based on single component systems, or systems of constant composition. These are a subset of the more general equations which can be derived, and which allow for changes in composition. It will be shown in Chapter 12 that if a change in composition occurs then another term defining the effect of this change is required. 6.1 The Maxwell relationships The concept of functional relationships between properties was introduced previously. For example, the Second Law states that, for a reversible process, T, s, u, p and v are related in the following manner T dS=dU+p dV (6.1) or, in specific (or molar) terms T ds=du+pdv (6.la) Rearranging eqn (6.la) enables the change of internal energy, du, to be written du= T ds-p dv (6.2) (6.2a) It will be shown in Chapter 12 that, in the general case where the composition can change, eqn (6.1) should be written TdS = dU +p dV - Cpidn, 1 where p, = chemical potential of component i n, = amount of component i (6.lb) The chemical potential terms will be omitted in the following analysis, although similar equations to those below can be derived by taking them into account.
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The Maxwell relationships 101 It can be seen from eqns (6.2) and (6.2a) that the specific internal energy can be represented by a three-dimensional surface based on the independent variables of entropy and specific volume. If this surface is continuous then the following relationships can be based on the mathematical properties of the surface. The restriction of a continuous surface means that it is ‘smooth’. It can be seen from Fig 6.1 that the p-v- T surface for water is continuous over most of the surface, but there are discontinuities at the saturated liquid and saturated vapour lines. Hence, the following relationships apply over the major regions of the surface, but not across the boundaries. For a continuous surface z = z(x, y) where z is a continuous function. Then dz= - dx+ - dy (6.3) (a (3 Let IV=($)~ and .=(:)I (6.4) Then dz = M dx + N dy (6.5) For continuous functions, the derivatives aZz d2Z ax ay ay ax -
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102 General thermodynamic relationships are equal, and hence (?)x=(:)y Consider also the expressions obtained when z = z(x, y) and x and y are themselves related to additional variables u and v, such that x = x(u, v) and y = y(u, v). Then Let z = v, and u = x, then x = x(z) and (2) V =0, and ($)v=l Hence giving (6.7) These expressions will now be used to consider relationships derived previously. The du=Tds-pdv (6.10) dh = T ds + v dp (6.1 1) df = -p dv - s dT (6.12) dg = - s (6.13) following functional relationships have already been obtained Consider the expression for du, given in eqn (6.10) then, by analogy with eqn (6.3) T = ( $)v, -p = (%), and (5). = -( $)" (6.14) In a similar manner the following relationships can be obtained, for constant composition or single component systems T=($); ($)s=($) P -P=(:); -s=($; ($)v=($)T v=($); -.=($); ($)p=($)T (6.15) (6.16) (6.17)
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The Maxwell relationships 103 If the pairs of relationships for T are equated then (z)v=($)p and, shilarly In addition to these equivalences, eqns (6.14) to (6.17) also show that ($l= ($)p = -MT Equations (6.19) are called the Maxwell relationships.
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This note was uploaded on 03/09/2010 for the course MECHANICAL ME9802701 taught by Professor Prof.william during the Spring '10 term at Institut Teknologi Bandung.

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06 - G eneral Thermodynamic Relationships single component systems o r systems of constant com position The relationships which follow are based on

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