The chemical potential (μ) of a 2 molar Na2SO4 solution is expressed in terms of mean ionic activity co - efficient (γ±) as
μo + 5RTIn2 + 3RTln γ±
Chemical Potential Explained
The chemical potential (μ) of a substance in a solution represents its contribution to the total Gibbs free energy of the solution. For a solute in a non-ideal solution, the chemical potential is given by the equation:
$$ \mu = \mu^\circ + RT \ln a $$where:
- $$\mu^\circ$$ is the standard chemical potential
- $$R$$ is the ideal gas constant
- $$T$$ is the absolute temperature
- $$a$$ is the activity of the solute
The activity ($$a$$) accounts for the non-ideal behavior of the solute particles in the solution.
Na2SO4 Dissociation in Solution
Sodium sulfate ($\text{Na}_2\text{SO}_4$) is a strong electrolyte. When it dissolves in water, it dissociates completely into its constituent ions:
$$ \text{Na}_2\text{SO}_4 \rightarrow 2\text{Na}^+ + \text{SO}_4^{2-} $$This dissociation shows that for every one formula unit of $\text{Na}_2\text{SO}_4$, we get:
- Two sodium cations ($\text{Na}^+$), so the number of cations $$v_+ = 2$$.
- One sulfate anion ($\text{SO}_4^{2-}$), so the number of anions $$v_- = 1$$.
- The total number of ions is $$v = v_+ + v_- = 2 + 1 = 3$$.
Activity of Electrolyte Solution
For a strong electrolyte that dissociates into $$v_+$$ cations and $$v_-$$ anions, the activity ($$a$$) is related to the concentrations and activity coefficients of the individual ions. The activity can be expressed in terms of the mean ionic activity coefficient ($\gamma_{\pm}$):
$$ a = \left(\frac{C_+}{C^\circ}\right)^{v_+} \left(\frac{C_-}{C^\circ}\right)^{v_-} \gamma_{\pm}^v $$Here, $$C_+$$ and $$C_-$$ are the molar concentrations of the cation and anion, respectively, and $$C^\circ$$ is the standard state concentration (typically 1 M). Given the $\text{Na}_2\text{SO}_4$ solution has a concentration $$C = 2$$ M:
- The concentration of $\text{Na}^+$ ions is $$C_+ = v_+ \times C = 2 \times 2\text{ M} = 4\text{ M}$$.
- The concentration of $\text{SO}_4^{2-}$$ ions is $$C_- = v_- \times C = 1 \times 2\text{ M} = 2\text{ M}$$.
Assuming the standard state concentration $$C^\circ = 1$$ M, and using $$v_+ = 2$$, $$v_- = 1$$, $$v = 3$$, we substitute these values into the activity expression:
$$ a = \left(\frac{4}{1}\right)^2 \left(\frac{2}{1}\right)^1 \gamma_{\pm}^3 $$ $$ a = (4)^2 \cdot (2)^1 \cdot \gamma_{\pm}^3 $$ $$ a = 16 \cdot 2 \cdot \gamma_{\pm}^3 $$ $$ a = 32 \gamma_{\pm}^3 $$Calculating Chemical Potential Expression
Now substitute the calculated activity ($$a = 32 \gamma_{\pm}^3$$) back into the chemical potential equation:
$$ \mu = \mu^\circ + RT \ln a $$ $$ \mu = \mu^\circ + RT \ln (32 \gamma_{\pm}^3) $$Using the properties of logarithms, $$\ln(xy) = \ln x + \ln y$$ and $$\ln(x^p) = p \ln x$$, we can expand the logarithmic term:
$$ \mu = \mu^\circ + RT (\ln 32 + \ln \gamma_{\pm}^3) $$ $$ \mu = \mu^\circ + RT (\ln 32 + 3 \ln \gamma_{\pm}) $$Since $$32 = 2^5$$, we have $$\ln 32 = \ln 2^5 = 5 \ln 2$$. Substitute this into the expression:
$$ \mu = \mu^\circ + RT (5 \ln 2 + 3 \ln \gamma_{\pm}) $$Finally, distribute $$RT$$:
$$ \mu = \mu^\circ + 5 RT \ln 2 + 3 RT \ln \gamma_{\pm} $$This expression for the chemical potential matches one of the given options.
Important Questions from Electrochemistry
- At $298 \, K$, given the standard electrode potentials: $E^\circ_{Cu^{2+}/Cu} = 0.34 \, V$, $E^\circ_{Zn^{2+}/Zn} = -0.76 \, V$, $E^\circ_{Fe^{2+}/Fe} = -0.44 \, V$, and $E^\circ_{Ag^{+}/Ag} = 0.80 \, V$.
Based on these values, which of the following reactions is NOT expected to occur spontaneously under standard conditions? - Which of the following processes is required for extracting metal from cinnabar ore?
You are given three metals 'X', 'Y' and 'Z'. Metal 'X' is found to react with an aqueous solution of both YSO 4and ZSO 4whereas metal 'Z' is found to react only with aqueous solution of YSO 4. Based on these observations, select the correct statement from the following.
The mobility of a divalent cation in water is 8 × 10-8 m2 V-1 s-1. The effective radius of the ion is (viscosity of water = 1 cP : c = 1.6 × 10-19 C)
The electrical double layer model among the following that consists of both fixed and diffuse layers is
