- #1
Dario56
- 289
- 44
In the textbook Electrochemical Systems by Newman and Alyea, chapter 14: The definition of some thermodynamic functions, chemical potential of component (ionic or neutral) is written as a function of absolute activity: $$ \mu_i = RTln(\lambda_i) \tag {1} $$
where ##\lambda_i## is the absolute activity of the component ##i##.
What I know from thermodynamics is the following: $$ \mu_i = \mu_i ^⦵ + RTln \frac { f_i}{f_i ^⦵} = \mu_i ^⦵ + RT ln\frac {a_i}{a_i ^⦵} \tag{2}$$
where ##f_i## and ##f_i^⦵$## are partial and standard fugacities of component, respectively. It is important to note that ##a_i = \frac {f_i}{f_i^⦵}## and ##a_i ^⦵ = 1##.
Since we don't know the values of chemical potential, we can express them relatively to the standard state if we take that chemical potential at standard state is equal to zero: $$ \mu_i = RTln(a_i) = RTln(\lambda_i)\tag {3} $$
This is all well and good.
For mixtures in general (solutions of electrolytes are mixtures), standard state of the component is usually taken as a state of pure component at the temperature and pressure of the system (pure liquid for solvent or pure solid for solute). Choice of such standard state allows us to express chemical potential of the component in a mixture as a function of activity in a familiar way: $$ \mu_i = \mu_i ^⦵ + RT ln (x_i \gamma_i) \tag {4}$$
where ##\gamma_i## is the activity coefficient of the component ##i##. It is also evident that ##a_i = x_i \gamma_i##.
If solution is diluted than mole fractions are directly proportional to the molarity of the component ##m_i## (##m_i = \frac {x_i}{M(Solvent)})##
This allows us to express equation 5 in terms of molarity: $$\mu_i = \mu_i ^⦵ + RTln(m_i\gamma_i M(solvent)) \tag{5} $$
Standard state chemical potential is now redefined as we add ##RTln(M(solvent))## to its previous value and refers to the state of ideal solution with unit molarity: $$ \mu_i = \mu_i^{⦵'} + RTln(m_i \gamma_i) \tag{6} $$
Comparing with equation 2 we can write: $$ \frac {a_i}{a_i ^⦵} = \frac {\lambda_i}{\lambda_i ^⦵} = m_i \gamma_i \tag{7} $$
Next equation is written: $$ \lambda_i = m_i\gamma_i \lambda_i ^⦵ \tag {8} $$
In the textbook, it is explained that standard activity ##\lambda_i ^⦵## is a proportionality constant independent of composition and electrical state, but dependent on temperature, pressure and solute type. However, by definition of activity this value should always be equal to 1 and thus independent on any variable. Standard fugacity doesn't need to be equal to 1, but activity must be since ##\lambda_i ^⦵ = \frac {f_i^⦵}{f_i ^⦵}##, as far as my knowledge of thermodynamics goes.
where ##\lambda_i## is the absolute activity of the component ##i##.
What I know from thermodynamics is the following: $$ \mu_i = \mu_i ^⦵ + RTln \frac { f_i}{f_i ^⦵} = \mu_i ^⦵ + RT ln\frac {a_i}{a_i ^⦵} \tag{2}$$
where ##f_i## and ##f_i^⦵$## are partial and standard fugacities of component, respectively. It is important to note that ##a_i = \frac {f_i}{f_i^⦵}## and ##a_i ^⦵ = 1##.
Since we don't know the values of chemical potential, we can express them relatively to the standard state if we take that chemical potential at standard state is equal to zero: $$ \mu_i = RTln(a_i) = RTln(\lambda_i)\tag {3} $$
This is all well and good.
For mixtures in general (solutions of electrolytes are mixtures), standard state of the component is usually taken as a state of pure component at the temperature and pressure of the system (pure liquid for solvent or pure solid for solute). Choice of such standard state allows us to express chemical potential of the component in a mixture as a function of activity in a familiar way: $$ \mu_i = \mu_i ^⦵ + RT ln (x_i \gamma_i) \tag {4}$$
where ##\gamma_i## is the activity coefficient of the component ##i##. It is also evident that ##a_i = x_i \gamma_i##.
If solution is diluted than mole fractions are directly proportional to the molarity of the component ##m_i## (##m_i = \frac {x_i}{M(Solvent)})##
This allows us to express equation 5 in terms of molarity: $$\mu_i = \mu_i ^⦵ + RTln(m_i\gamma_i M(solvent)) \tag{5} $$
Standard state chemical potential is now redefined as we add ##RTln(M(solvent))## to its previous value and refers to the state of ideal solution with unit molarity: $$ \mu_i = \mu_i^{⦵'} + RTln(m_i \gamma_i) \tag{6} $$
Comparing with equation 2 we can write: $$ \frac {a_i}{a_i ^⦵} = \frac {\lambda_i}{\lambda_i ^⦵} = m_i \gamma_i \tag{7} $$
Next equation is written: $$ \lambda_i = m_i\gamma_i \lambda_i ^⦵ \tag {8} $$
In the textbook, it is explained that standard activity ##\lambda_i ^⦵## is a proportionality constant independent of composition and electrical state, but dependent on temperature, pressure and solute type. However, by definition of activity this value should always be equal to 1 and thus independent on any variable. Standard fugacity doesn't need to be equal to 1, but activity must be since ##\lambda_i ^⦵ = \frac {f_i^⦵}{f_i ^⦵}##, as far as my knowledge of thermodynamics goes.