# Definition of activity

1. Jun 15, 2017

### ussername

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. Jun 15, 2017

### BvU

$T$ is the current temperature. Lemma is pretty clear about that !

3. Jun 15, 2017

### ussername

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.

4. Jun 15, 2017

### BvU

Yes they certainly can.

5. Jun 16, 2017

### ussername

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:
$$\mu _{i}-\mu _{i}^{0}=RT\cdot \ln a_{i}$$

6. Jun 16, 2017

### BvU

What do you obtain ?

7. Jun 16, 2017

### ussername

In that case I have an integral:

$$\mu _{i}-\mu _{i}^{0}=R\cdot \int_{a_{i}^{0}}^{a_{i}} T(a_{i})\cdot d\ln a_{i}$$

8. Jun 16, 2017

### BvU

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
9. Jun 16, 2017

### ussername

That is not true. It that case μi should be always zero.

10. Jun 16, 2017

### BvU

No. It is a constant. So the $\ d(\mu _{i}^\ominus) = 0$.

11. Jun 16, 2017

### ussername

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).

12. Jun 16, 2017

### BvU

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.

13. Jun 16, 2017

### ussername

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:

$$d\mu_i = d\left ( RT\ln a_i \right )$$

14. Jun 16, 2017

### BvU

Why not ?

15. Jun 16, 2017

### ussername

Sorry I meant:
$$d\mu_i = RT\cdot d\ln a_i$$

16. Jun 16, 2017

### BvU

Is wrong. T is not a constant like R

17. Jun 16, 2017

### ussername

Yes the total differential should be probably:
$$d\mu _{i}(dT,dp,dx_{1},...,dx_{N})=RT\cdot d\ln a_{i}(dp,dx_{1},...,dx_{N})+R\cdot \ln a_{i}\cdot dT$$

18. Jun 22, 2017

### DrDu

Differentials aren't functions of differentials!

19. Jun 22, 2017

### DrDu

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 $a_i(T)$.

20. Jun 24, 2017

### ussername

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:
$$RT' \cdot \ln \left( a_i(T',p',T_{std}',p_{std}') \right) = RT \cdot \ln \left( a_i(T,p,T_{std},p_{std}) \right) - \int_{T}^{T'} \overline{S}_i\, dT + \int_{T_{std}}^{T_{std}'} \overline{S}_{std,i} \, dT + \int_{p}^{p'} \overline{V}_i\, dp - \int_{p_{std}}^{p_{std}'} \overline{V}_{std,i}\, dp$$