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Definition of activity

  1. Jun 15, 2017 #1
    The definition of activity is:
    $$\mu _{i}=\mu _{i}^{0}+RT\cdot \ln a_{i}$$
    where μi is the chemical potential of i in current state and μi0 is the chemical potential of i in standard state.
    The current and standard state have the same temperature or can their temperature differ?
    If their temperature can differ, than T in the definition equation is the temperature of current state?
     
  2. jcsd
  3. Jun 15, 2017 #2

    BvU

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    ##T## is the current temperature. Lemma is pretty clear about that !
     
  4. Jun 15, 2017 #3
    But I'm asking if the temperature of current state and standard state can differ. Probably yes but it is not mentioned within the definition on wikipedia.
     
  5. Jun 15, 2017 #4

    BvU

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    Yes they certainly can.
     
  6. Jun 16, 2017 #5
    In such a case this definition is not equivalent to the definition of activity:
    $$d\mu _{i}=RT\cdot d\ln a_{i}$$ $$a_{i}^{0}=1$$
    because when integrating with changing temperature, I generally do not obtain this equation:
    $$\mu _{i}-\mu _{i}^{0}=RT\cdot \ln a_{i}$$
     
  7. Jun 16, 2017 #6

    BvU

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    What do you obtain ?
     
  8. Jun 16, 2017 #7
    In that case I have an integral:

    $$\mu _{i}-\mu _{i}^{0}=R\cdot \int_{a_{i}^{0}}^{a_{i}} T(a_{i})\cdot d\ln a_{i}$$
     
  9. Jun 16, 2017 #8

    BvU

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    I don't recognize $$ d\mu _{i}=RT\cdot d\ln a_{i}$$ as a definition. ##a## is not an independent variable, so ##T(a)## seems weird.

    The expression is kind of a tautology when I substitute the definition : ##\ d\mu _{i} = d(\mu _{i} - \mu _{i}^\ominus)##

    Check here under activity and activity coefficients

    Alternatively we can consult @Chestermiller who might well have a didactically more responsible answer
     
    Last edited: Jun 16, 2017
  10. Jun 16, 2017 #9
    That is not true. It that case μi should be always zero.
     
  11. Jun 16, 2017 #10

    BvU

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    No. It is a constant. So the ## \ d(\mu _{i}^\ominus) = 0 ##.
     
  12. Jun 16, 2017 #11
    Now I don't know what you wanted to show, but i is the infinitesimal change of chemical potential of i during the infinitesimal process (eg. addition of dni).
     
  13. Jun 16, 2017 #12

    BvU

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    And here's me thinking $$ \mu_i = \left ( \partial G\over \partial N_i \right )_{T,P,N_{j\ne i}}$$ (fortunately some others seem to think so too).

    What I showed is that ##d\mu_i = d\left ( RT\ln a_i \right ) \ ## boils down to ##d\mu_i = d\mu_i \ ## if you insert ##\ a_i = e^{\mu_i-\mu^\ominus_i\over RT}\ ##, not very surprising and not very interesting.

    If you define $$a_i \equiv e^{\mu_i-\mu^\ominus_i\over RT}$$ then trivially and without integrating, just taking logarithms: $$\mu_i=\mu^\ominus_i+ RT\ln a_i $$ And we could have ended this thread after post #2.
     
  14. Jun 16, 2017 #13
    What I wanted to say:
    if we take the definition:
    $$\mu_i=\mu^\ominus_i+ RT\ln a_i $$
    for arbitrary ##T,T^\ominus##, than this is not valid:

    $$d\mu_i = d\left ( RT\ln a_i \right ) $$
     
  15. Jun 16, 2017 #14

    BvU

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    Why not ?
     
  16. Jun 16, 2017 #15
    Sorry I meant:
    $$d\mu_i = RT\cdot d\ln a_i $$
     
  17. Jun 16, 2017 #16

    BvU

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    Is wrong. T is not a constant like R
     
  18. Jun 16, 2017 #17
    Yes the total differential should be probably:
    $$d\mu _{i}(dT,dp,dx_{1},...,dx_{N})=RT\cdot d\ln a_{i}(dp,dx_{1},...,dx_{N})+R\cdot \ln a_{i}\cdot dT$$
     
  19. Jun 22, 2017 #18

    DrDu

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    Differentials aren't functions of differentials!
     
  20. Jun 22, 2017 #19

    DrDu

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    For fixed concentrations ##d \mu_i=-s_i dT +v_i dP## where ##s_i## and ##v_i## are molar entropy and volume, respectively.
    A similar equation holds for the standard chemical potential.
    ##d \mu^\ominus_i=-s^\ominus_i dT +v^\ominus_i dP##, with the standard molar entropy and volume.
    So if you change T, you get
    ##\mu(T)=\mu^\ominus(T) +RT_0 \ln a_i(T_0) -(\int_{T_0}^T (s_i(T')-s^\ominus(T')) dT')= \mu^\ominus(T) +RT \ln a_i(T)## which you may easily solve for ##a_i(T)##.
     
  21. Jun 24, 2017 #20
    Also the change of chemical potential with temperature resp. pressure is:
    $$\left( \frac{\partial \mu_i}{\partial T} \right)_{p,\vec{n}} = - \overline{S}_i$$ $$\left( \frac{\partial \mu_i}{\partial p} \right)_{T,\vec{n}} = \overline{V}_i$$
    Also the chemical activity can be defined for current state and standard state with arbitrary temperature and pressure:
    $$\mu_i(T,p) \equiv \mu_i^0(T^0,p^0) + RT \cdot \ln \left( a_i(T,p,T^0,p^0) \right) $$

    And the correction of chemical activity with changing temperature and pressure of current resp. standard state is:
    $$RT' \cdot \ln \left( a_i(T',p',T_{std}',p_{std}') \right) = RT \cdot \ln \left( a_i(T,p,T_{std},p_{std}) \right) - \int_{T}^{T'} \overline{S}_i\, dT + \int_{T_{std}}^{T_{std}'} \overline{S}_{std,i} \, dT + \int_{p}^{p'} \overline{V}_i\, dp - \int_{p_{std}}^{p_{std}'} \overline{V}_{std,i}\, dp$$
     
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