# chemical potential derivation

(i.e. The values of $$\mu$$ and $$\mu\st$$ include the molar internal energy whose absolute value can only be calculated from the Einstein relation; see Sec. N �>T�%u� 7.8.16 becomes \begin{equation} \ln\phi \approx \frac{Bp}{RT} \tag{7.8.18} \end{equation}. , at constant temperature : G A #soln = µ A = µ o A + RT ln X A, n A. G B #soln = µ B = µ o B + RT ln X B, etc. chemical potential, So far we have "Chemical Potential Dependence on Temperature and Pressure", http://demonstrations.wolfram.com/ChemicalPotentialDependenceOnTemperatureAndPressure/, Chemical Potential Dependence on Temperature and Pressure. )\) Think of a beam balance and you get the drift. And since systems tend to seek a minimum aggregate Gibbs function, the chemical potential will point to the direction the system can move in order to reduce the total Gibbs function. in phase. U 1 Point A is the gas standard state. It is useful to discern between the internal and the external chemical potential.  $$\newcommand{\mB}{_{\text{m},\text{B}}} % subscript m,B (m=molar)$$ The chemical potential of A (in liquid and in the vapour) is given by lot of relations between the partial derivatives of state functions and ignore it (if we don't, math will do it for us as as soon as we write down at. Or, by substituting the definition for the chemical potential, and evaluating the pressure and temperature derivatives as was done in Chapter 6: But as it turns out, the chemical potential can be defined as the partial molar derivative any of the four major thermodynamic functions $$U$$, $$H$$, $$A$$, or $$G$$: The last definition, in which the chemical potential is defined as the partial molar Gibbs function is the most commonly used, and perhaps the most useful (Equation \ref{eq1}). 7.8.6: \begin{gather} \s{ \mu = \mu\st\gas + RT \ln \frac{\fug}{p\st}} \tag{7.8.7} \cond{(pure gas)} \end{gather} or \begin{gather} \s{\fug \defn p\st \exp\left[ \frac{\mu-\mu\st\gas }{RT} \right]} \tag{7.8.8} \cond{(pure gas)} \end{gather} Note that fugacity has the dimensions of pressure. 7.8.5: $$\mu(p') - \mu\st = \int_{p\st}^{p'} (RT/p) \difp = RT\ln(p'/p\st)$$. This is one way of writing down the $$\newcommand{\tx}{\text{#1}} % text in math mode$$  $$\newcommand{\difp}{\dif\hspace{0.05em} p} % dp$$ In other words, the "chemical potential μ " is a measure of how much the free enthalpy (or the free energy) of a system changes (by dGi) if you add or remove a number dni particles of the particle species i while keeping the number of the other particles (and the temperature T and the pressure p) constant: d Gi. no problem. 0  >> %PDF-1.5 The calculating of excess chemical potential is not limited to homogeneous systems, but has also been extended to inhomogeneous systems by the Widom insertion method, or other ensembles such as NPT and NVE.  $$\newcommand{\irr}{\subs{irr}} % irreversible$$ endobj  $$\newcommand{\solmB}{\tx{(sol,\,m\B)}}$$ Unless otherwise noted, LibreTexts content is licensed by CC BY-NC-SA 3.0. The partial free energy per mole of a component is its chemical potential, so we may write: G A #soln = n A µ A, G B #soln = n B µ B, etc.  $$\newcommand{\rxn}{\tx{(rxn)}}$$ As the partial most Gibbs function, it is easy to show that, where $$V$$ is the molar volume, and $$S$$ is the molar entropy.  $$\newcommand{\mol}{\units{mol}} % mole$$ We call it the chemical potential: If two systems are at the same temperature and only have a single chemical species and the same value of the chemical potential, there is no net ux of particles from one side to another. Chemical potential as a function of pressure is also shown for the solid-liquid phase change for ethanol, which has a different pressure dependence than water. (  $$\newcommand{\gpht}{\small\gph} % gamma phase tiny superscript$$, $$\newcommand{\dif}{\mathop{}\!\mathrm{d}} % roman d in math mode, preceded by space$$ The result is … Have questions or comments? i non-ideal gas. Also, note that the axis scale is not the same for all plots so that the differences in chemical potential are easier to see. In full generality, we have two equations: We now must change the particle xڅ�;O�0����-��{^?�ם�DE:�H.��8����M ��jF#}3�����G/�f�'�Q�5�=��.h�ԢlK4��:ŷt��\h��;ʱ9ǉ��'�q3�4k�Į�VN,W���#�y�O�mYv@��ZU�[�d4�Q� _Lc\jU��������f�ӺƯP Check "add salt" to see the effect of adding salt to liquid water, and set the salt concentration with a slider. We can also turn it around: Vacancies in For the atoms in the lower volume, ml has a similar structure, but the gravitational potential energy is zero. In cell membranes, the electrochemical potential is the sum of the chemical potential and the membrane potential. one fell swoop we also include liquids in this). + 5 2 # (4) The next step, however, is a bit more problematic. standard state to be, In other words, the standard reference state is Q In a binary mixture , if one partial molar quantity increases the other must decrease. V V /Length 2837 (st.4) and (st.5), provided we know how particle energy depends on its momentum, i.e. {\displaystyle Q(N,V,T)={\frac {V^{N}}{\Lambda ^{dN}N! We can derive a similar equation for any other partial molar quantity. μ {\displaystyle V}  $$\newcommand{\timesten}{\mbox{\,\times\,10^{#1}}}$$  $$\newcommand{\Rsix}{8.31447\units{J\,K\per\,mol\per}} % gas constant value - 6 sig figs$$, $$\newcommand{\jn}{\hspace3pt\lower.3ex{\Rule{.6pt}{2ex}{0ex}}\hspace3pt}$$ potential. T The reason is that people with a mostly, First we will address, somewhat  $$\newcommand{\Pd}{\left( \dfrac {\partial #1} {\partial #2}\right)_{#3}} % Pd{}{}{} - Partial derivative, built-up$$ ) The LibreTexts libraries are Powered by MindTouch® and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products. 1 If we  $$\newcommand{\sln}{\tx{(sln)}}$$  $$\newcommand{\liquid}{\tx{(l)}}$$ L Unfortunately, there is another drawback. exp 5.5) \begin{gather} \s{ \mu \defn G\m = \frac{G}{n} } \tag{7.8.1} \cond{(pure substance)} \end{gather} That is, $$\mu$$ is equal to the molar Gibbs energy of the substance at a given temperature and pressure.

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