Colloid chemistry chapter 4 thermodynamics of surface edit

COLLOID CHEMISTRY Chapter – Thermodynamics of Surface Dr Ngo Thanh An Surface excess • The presence of an interface influences generally all thermodynamic parameters of a system • To consider the thermodynamic of a system with an interface, we divide that system into three parts: The two bulk phases with volumes Vα and Vβ and the interface σ Surface excess Gibbs dividing plane • In Gibbs convention the two phases are thought to be separated by an infinitesimal thin boundary layer, the Gibbs dividing plane (this is of course an idealization) • Gibbs dividing plane also called an ideal interface • In Gibbs model the interface is ideally thin (V σ = 0) and the total volume is V = Vα + Vβ Surface excess Gibbs dividing plane ∞ 𝑧𝑒 𝑧𝑒 Γ =∫ [ 𝜌 ( 𝑧 ) − 𝜌 𝑏 ] 𝑑𝑧 −∫ [ 𝜌 𝑎 − 𝜌 ( 𝑧 ) ] 𝑑𝑧 =0 • In Gibbs convention the two phases α and β are separated by an ideal interface σ which is infinitely thin: Guggenheim explicitly treated an extended interphase with a volume Surface excess Interfacial excess N i  N i  ciV   ci V   i N i  A V  V  V   i  i   i  i  N Ni  c V  c  c V  Surface excess    c V  c        V N  N1  c V  c  c V  N N Multiply both sides of equation (1) by c  (1)  (2)  c2  c2        c1  c1             c2  c2 c2  c2 c2  c2      N1     c1 V    N1          c1  c1   c1  c1   c1  c1     c2  c2       c1  c1 V      c1  c1    (3) Surface excess Equation (2) – equation (3)  c2  c2       N  N    N  c V   c  c  V 2 2   c  c    c2  c2    c2  c2    c1 V     N1        c1  c1   c1  c1     c2  c2    c  c V      c1  c1              c  c c  c     2 2   N  c2 V   N1  c1 V    N  N1        c1  c1   c1  c1            c  c c  c   i   i i i N i  N1   N  c V  ( N  c V ) i i 1 c1  c1 c1  c1 Surface excess Finally, dividing through by the interfacial area A, gives:   c2  c2    N 2 N1  c2  c2        N  c2 V   N1  c1 V            A A  c1  c1  A   c1  c1                N 2 N1  c2  c2   c  c   2    N  c2 V   N1  c1 V            A A  c1  c1  A   c1  c1        2(1)   c  c 2  1 2 1 c1  c1 i(1)   c  c i  1 i i c1  c1  Surface excess Interfacial excess • The right side of the equation does not depend on the position of the Gibbs dividing plane and thus, also the left side is invariant We divide this quantity by the surface area and obtain the invariant quantity i(1)   c  c i  1 i i c1  c1  It is called relative adsorption of component i with respect to component  This is an important quantity because it can be determined experimentally and it can be measured by determinig the surface tension of liquid versus the concentration of the solute Fundamentals of thermodynamic relations a Internal energy and Helmholtz energy  V V  V  dV dV   dV   dV dV  dV U U   U   U   i  i N i N  N  N   S S  S  S   i  Fundamentals of thermodynamic relations d Free surface energy dU  TdS    i dN i  dA U  TS    i N i  A F A   i N   i dF   S  dT   i dN i  dA S   A T , N i   T A, N i Fundamentals of thermodynamic relations e Eurler’s theorem A two-variable homogenous function: f = f(x,y) Exact differential for f: Integrate both sides of equation: Thefore, we have: Fundamentals of thermodynamic relations e Eurler’s theorem Application of Euler’s theorem On the interface, we have: Integrate both sides of the above equation: