Electrochemistry

The relation between molar conductivity and concentration for strong electrolytes is given by

$\Lambda_M = \Lambda_M^0 - b\sqrt{c}$

For two concentrations $ c_1 $ and $ c_2 $, the equation becomes:

$\Lambda_{M,1} - \Lambda_{M,2} = b(\sqrt{c_2} - \sqrt{c_1})$

Therefore,

$b = \frac{\Lambda_{M,1} - \Lambda_{M,2}}{\sqrt{c_2} - \sqrt{c_1}} = \frac{\Lambda_{M,1} - \Lambda_{M,2}}{0.1}$

For NaNO₃:

$b = \frac{111 - 101}{0.1} = 100$

So the limiting molar conductivity is:

$[\Lambda_M^0]_{\mathrm{NaNO_3}} = \Lambda_M + b\sqrt{c} = 111 + 100(0.1) = 121\, \mathrm{S\,cm^2/mol}$

For NaCl:

$b = \frac{117 - 107}{0.1} = 100$

$[\Lambda_M^0]_{\mathrm{NaCl}} = 117 + 100(0.1) = 127\, \mathrm{S\,cm^2/mol}$

For AgNO₃:

$b = \frac{125 - 116}{0.1} = 90$

$[\Lambda_M^0]_{\mathrm{AgNO_3}} = 125 + 90(0.1) = 134\, \mathrm{S\,cm^2/mol}$

Finding limiting molar conductivity of AgCl:

Using Kohlrausch’s law of independent migration of ions:

$[\Lambda_M^0]_{\mathrm{AgCl}} = [\Lambda_M^0]_{\mathrm{AgNO_3}} + [\Lambda_M^0]_{\mathrm{NaCl}} - [\Lambda_M^0]_{\mathrm{NaNO_3}}$

$= 134 + 127 - 121 = 140\, \mathrm{S\,cm^2/mol}$

Now, to find solubility (X):

The conductivity of the saturated solution is $ k = 1.40 \times 10^{-6}\, \mathrm{S\,cm^{-1}} $.

Since the electrolytic solution is very dilute,

$\Lambda_M = \Lambda_M^0 = \frac{k \times 1000}{c}$

So,

$c = \frac{k \times 1000}{\Lambda_M^0} = \frac{1.4\times10^{-6}\times10^3}{140} = 10^{-5}\, \mathrm{M}$

Thus, solubility $ X = 10^{-5} \, \mathrm{mol\,L^{-1}} $.

Finally,

$\log(X^{-1}) = \log(10^{-5})^{-1} = 5$

To determine the correct statement(s) regarding the electrochemical oxidation of hydrazine ($\mathrm{N}_2 \mathrm{H}_4$) by $\mathrm{O}_2$, let's analyze each option in detail.

Option A: $\mathrm{OH}^{-}$ ions react with $\mathrm{N}_2 \mathrm{H}_4$ at the anode to form $\mathrm{N}_2$$(\mathrm{~g})$ and water, releasing 4 electrons to the anode. This statement is accurate because in an electrochemical cell, the anode is where oxidation occurs. Hydrazine can be oxidized in the presence of $\mathrm{OH}^{-}$ ions to yield nitrogen gas and water, releasing electrons as a part of this redox reaction:

Option B: At the cathode, $\mathrm{N}_2 \mathrm{H}_4$ breaks to $\mathrm{N}_2$ $(\mathrm{~g})$ and nascent hydrogen released at the electrode reacts with oxygen to form water. This statement is incorrect because, usually, the cathode is where reduction occurs, not a breaking down of hydrazine. Instead, reduction reactions involve the gain of electrons. Additionally, at the cathode, it's more common for oxygen to be reduced rather than hydrazine itself breaking down.

Option C: At the cathode, molecular oxygen gets converted to $\mathrm{OH}^{-}$. This statement is correct. At the cathode in an alkaline medium, oxygen undergoes reduction to form hydroxide ions $\mathrm{OH}^{-}$, as depicted by the half-reaction:

Option D: Oxides of nitrogen are major by-products of the electrochemical process. This statement is incorrect because the primary product from the electrochemical reaction of hydrazine is nitrogen gas ($\mathrm{N}_2$), and not oxides of nitrogen. The formation of oxides of nitrogen would be notable only under different specific conditions not mentioned in this context.

Therefore, the correct statements regarding the electrochemical oxidation of hydrazine by $\mathrm{O}_2$ are:

Option A and Option C.

Option (P) :

On adding $\mathrm{AgNO}_3$ solution to $\mathrm{KCl}$ solution precipitation of $\mathrm{AgCl}$ will occur due to which $\mathrm{Cl}^{-}$already present will be replaced by $\mathrm{NO}_3^{-}$ions. So conductance of solution will decrease till equivalence point. After complete precipitation of $\mathrm{AgCl}$, further added $\mathrm{AgNO}_3$ will increase the number of ions in resulting solution so conductance will increase.

Option (Q) :

On adding $\mathrm{KCl}$ solution to $\mathrm{AgNO}_3$ solution precipitation of $\mathrm{AgCl}$ will occur due to which already present $\mathrm{Ag}^{+}$ions will be replaced by $\mathrm{K}^{+}$ions in solution. So conductance of solution will increase. After complete precipitation of $\mathrm{AgCl}$ further added $\mathrm{KCl}$ will increase the number of ions in resulting solution so conductance will increase further.

Option (R) :

On adding $\mathrm{HCl}$ solution to $\mathrm{NaOH}$ solution, $\mathrm{OH}^{-}$will be replaced by $\mathrm{Cl}^{-}$ions so conductance of solution decreases. After complete neutralisation further added $\mathrm{HCl}$ will increase number of ions in the solution. So conductance will increase futher.

Option (S) :

On adding $\mathrm{CH}_3 \mathrm{COOH}$ solution to $\mathrm{NaOH}$ solution $\mathrm{OH}^{-}$will be replaced by $\mathrm{CH}_3 \mathrm{COO}^{-}$ions, so conductance of solution decreases. After complete neutralisation further added $\mathrm{CH}_3 \mathrm{COOH}$ will remain undissociated because it is a weak acid and there is also common ion effect on acetate ions. So number of ions in solution will remain almost constant therefore conductance of solution will remain constant.

The reaction involved is

$Zn(s) + C{u^{2 + }}(aq)$ $\rightleftharpoons$ $Z{n^{2 + }}(aq) + Cu(s)$

$\Delta G = \Delta G^\circ + 2.303RT\log {{[Z{n^{2 + }}]} \over {[C{u^{2 + }}]}}$

$\Delta G = - nFE^\circ + 2.303RT\log {{[Z{n^{2 + }}]} \over {[C{u^{2 + }}]}}$ ........ (1)

Substituting n = 2, E$^\circ$ = 1.1 and $[Z{n^{2 + }}] = 10$ and $[C{u^{2 + }}] = 1$ in Eq. (1), we get

$\Delta G = ( - 2 \times F \times 1.1) + 2.303RT\log {{10} \over 1}$

= 2.303 RT $-$ 2.2 F

For the given electrochemical cell, the half-cell reactions are

$\bullet$ At anode : H₂(g) $\rightleftharpoons$ 2H⁺(aq) + 2e^$-$

$\bullet$ At cathode : M⁴⁺(aq) + 2e^$-$ $\rightleftharpoons$ M²⁺(aq).

The overall cell reaction is

M⁴⁺(aq) + H₂(g) $\to$ M²⁺(aq) + 2H⁺(aq)

From Nernst equation, we have

${E_{cell}} = E_{cell}^o - {{2.303RT} \over {nF}}\log {{[{M^{2 + }}]{{[{H^ + }]}^2}} \over {[{M^{4 + }}]{p_{{H_2}}}}}$

Substituting the given values, we get

${E_{cell}} = (E_{{M^{4 + }}/{M^{2 + }}}^o - E_{{H^ + }/{H_2}}^o){{0.059} \over 2}\log {{[{M^{2 + }}]{{[{H^ + }]}^2}} \over {[{M^{4 + }}]{p_{{H_2}}}}}$

$0.092 = (0.151 - 0) - {{0.059} \over 2}\log {10^x}$

$0.092 = 0.151 - 0.0245\log {10^x}$

$ - 0.059 = - 0.0245x \Rightarrow x = 2$

In a galvanic cell, also known as a voltaic cell, a salt bridge plays a crucial role in maintaining electrical neutrality within the internal circuit, which is critical for the ongoing electrochemical reaction and thus the production of electrical energy. Here, we will analyze each of the given options in the context of the function of a salt bridge in a galvanic cell:

Option A: does not participate chemically in the cell reaction.

This statement is correct. The salt bridge in a galvanic cell does not directly participate in the chemical reactions occurring at the electrodes. Its primary function is to complete the electrical circuit between the cathode and anode by allowing the transfer of ions. The salt bridge contains a salt solution (usually KNO₃, KCl, or NH₄NO₃), where the ions migrate to oppose and balance the charge buildup due to the migration of electrons through the external circuit.

Option B: stops the diffusion of ions from one electrode to another.

This option can be misleading. The salt bridge does not stop the diffusion of ions from diffusing from one electrode to another but rather provides a pathway for ions to flow back and forth, which is essential to maintain the charge balance. Hence, this statement is not the best descriptor of the salt bridge's function.

Option C: is necessary for the occurrence of the cell reaction.

This option is correct. A salt bridge is necessary for the occurrence of the cell reaction because it maintains electrical neutrality in the electrochemical cell. Without the salt bridge, the flow of electrons through the external circuit would soon cease as the solutions in the anode and cathode compartments become respectively positively and negatively charged, which would stop the electrochemical reaction.

Option D: ensures mixing of the two electrolytic solutions.

This statement is incorrect. The purpose of the salt bridge is not to mix the two electrolytic solutions; in fact, it prevents them from mixing, which could otherwise result in a direct neutralization reaction that could interfere with the proper functioning of the cell. The ions in the salt bridge only migrate enough to balance the charges in the separate solutions.

In summary, the correct answers are Option A ("does not participate chemically in the cell reaction") and Option C ("is necessary for the occurrence of the cell reaction").

First, we will calculate the potential values needed for each of the queries in List – I based on the equations and the given standard reduction potential data:

For P. $E^o (\text{Fe}^{3+}, \text{Fe})$: This reaction involves a combination of two reactions:

$ \text{Fe}^{3+} + e^- \to \text{Fe}^{2+} \quad \text{(E}^o = +0.77 \text{ V)} $

$ \text{Fe}^{2+} + 2e^- \to \text{Fe} \quad \text{(E}^o = -0.44 \text{ V)} $

Using the formula to combine potentials for reactions in series, we get:

$ E^o_{\text{combined}} = E^o (\text{Fe}^{3+} \to \text{Fe}^{2+}) + E^o (\text{Fe}^{2+} \to \text{Fe}) = 0.77 \text{ V} - 0.44 \text{ V} = 0.33 \text{ V} $

For Q. $E^o (4\text{H}_2\text{O} \leftrightharpoons 4\text{H}^+ + 4\text{OH}^-)$:

By adding the two given reactions involving O₂ and water, we can calculate this potential:

$ 2 \text{H}_2\text{O} + 2 e^- \to \text{H}_2 + 2 \text{OH}^- \quad (2 \times +0.40 \text{ V}) $

$ \text{O}_2 + 2 \text{H}_2\text{O} + 4 e^- \to 4 \text{OH}^- \quad \text{(E}^o = +0.40 \text{ V)} $

Given that this is essentially the decomposition of water into its ions, the neuronal and overall reaction becomes:

$ 4 \text{H}_2\text{O} \leftrightharpoons 4 \text{H}^+ + 4 \text{OH}^- $

The E^o for this is effectively zero as it is a net reaction of water decomposing and reforming. Hence, the standard value of this is approximately -0.83 V (taking into account the sum of the reactions).

For R. $E^o (\text{Cu}^{2+} + \text{Cu} \to 2\text{Cu}^+)$:

Recall that:

$ \text{Cu}^{2+} + e^- \to \text{Cu}^+ \quad \text{(E}^o = +0.16 \text{ V)} $

$ \text{Cu}^+ + e^- \to \text{Cu} \quad \text{(E}^o = +0.52 \text{ V)} $

To find $E^o (\text{Cu}^{2+} + \text{Cu} \to 2\text{Cu}^+),$ we have to reverse the second reaction and combine it with the first:

$ E^o = 0.16 \text{ V} - 0.52 \text{ V} = -0.36 \text{ V} $

For S. $E^o (\text{Cr}^{3+}, \text{Cr}^{2+})$:

Directly taken from the data:

$ \text{Cr}^{3+} + e^- \to \text{Cr}^{2+} \quad \text{(E}^o = -0.74 \text{ V)} $

Now we match these with the correct values:

P. 0.33 V (No exact match, but since this value should be positive and closest to zero among the choices, associate with least negative value)

Q. -0.83 V

R. -0.36 V

S. -0.74 V

Finally, comparing these calculated values to the options, we find:

Option D (P - 3; Q - 4; R - 1; S - 2) matches the calculated results.

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(P) The weak acid (Y) is partially dissociated as follows:

$ \mathrm{CH}_3 \mathrm{COOH}(aq) \rightleftharpoons \mathrm{CH}_3 \mathrm{COO}^{-}(aq) + \mathrm{H}^{+}(aq) $

(a) Adding the base triethylamine will result in its protonation by $\mathrm{H}^{+}$. As more $\mathrm{H}^{+}$ ions are consumed, the reaction shifts forward, producing $\mathrm{CH}_3 \mathrm{COO}^{-}$ ions and protonated amine, increasing the solution's conductivity.

$ (\mathrm{C}_2 \mathrm{H}_5)_3 \ddot{\mathrm{N}} + \mathrm{H}^{+} \rightarrow (\mathrm{C}_2 \mathrm{H}_5)_3 \mathrm{NH}^{+} $

Net equation:

$ \mathrm{CH}_3 \mathrm{COOH}(aq) + (\mathrm{C}_2 \mathrm{H}_5)_3 \ddot{\mathrm{N}} \rightarrow (\mathrm{C}_2 \mathrm{H}_5)_3 \stackrel{+}{\mathrm{N}}\mathrm{H} + \mathrm{CH}_3 \mathrm{COO}^{-} $

(b) Once all the acid is neutralized, adding more $(\mathrm{C}_2 \mathrm{H}_5)_3 \mathrm{N}$ will not further affect the solution's conductivity.

Hence, initially conductivity increases and then remains constant.

Option (P) in List-I matches with option 3 in List-II.

(Q) The reaction between potassium iodide (KI) and silver nitrate $(\mathrm{AgNO}_3)$ occurs as follows:

$ \mathrm{KI}(0.1 \, \mathrm{M}) + \mathrm{AgNO}_3(0.01 \, \mathrm{M}) \rightarrow \mathrm{KNO}_3 + \mathrm{AgI}_{(\mathrm{ppt})} $

(a) As KI is added to $\mathrm{AgNO}_3$, silver iodide precipitates, and conductivity due to $\mathrm{AgNO}_3$ decreases, but potassium nitrate formation increases the solution's conductivity. These effects balance out, so conductivity does not change much initially.

(b) After all $\mathrm{AgNO}_3$ is consumed, adding more KI, a strong electrolyte, will increase the solution's conductivity.

Hence, initially conductivity does not change much and then increases.

(R) The weak acid $\mathrm{CH}_3 \mathrm{COOH}$ dissociates as follows:

$ \mathrm{CH}_3 \mathrm{COOH}(aq) \rightleftharpoons \mathrm{CH}_3 \mathrm{COO}^{-}(aq) + \mathrm{H}^{+}(aq) $

When it is added to potassium hydroxide (KOH), a strong electrolyte, the following reaction takes place:

$ \mathrm{CH}_3 \mathrm{COOH} + \mathrm{KOH} \rightarrow \mathrm{CH}_3 \mathrm{COO}^{-} \mathrm{K}^{+} + \mathrm{H}_2 \mathrm{O} $

The potassium ion (K$^+$) from KOH, which has a higher migration velocity, is replaced by the hydrogen and acetate ion $\mathrm{CH}_3 \mathrm{COO}^{-}$, which has lower migration velocity. Consequently, conductivity decreases. After all KOH is neutralized by the weak acetic acid $\mathrm{CH}_3 \mathrm{COOH}$, adding more acid will increase conductivity only slightly due to its lower degree of dissociation. Hence, initially the conductivity decreases but then changes very little.

(S) Hydrogen iodide (HI) is a strong acid and completely dissociates into hydrogen and iodide ions. Addition of sodium hydroxide (NaOH) forms sodium iodide and water. The hydrogen ions, which have high migration velocity, are replaced by slower-moving sodium ions, causing a decrease in conductivity. After all the HI is neutralized, conductivity increases with further addition of NaOH due to the increased concentration of $\mathrm{OH}^{-}$ ions.

P - 3; Q - 4; R - 2; S - 1

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To find the solubility product (K_sp) for the sparingly soluble salt, MX₂, in a concentration cell setup, we first analyze how the emf is related to the concentration differences across the cell. The reaction at each electrode involves the metal ion M²⁺ and the metal M, whereas the net cell reaction has no change in number of moles on both sides of the equation due to the symmetry of the cell. Thus, the notation for the cell reaction is:

$ M(s) \longleftrightarrow M^{2+}(aq) + 2e^{-} $

Given that it's a concentration cell, the emf generated is due to the concentration difference of the M²⁺ ions at the two electrodes. The emf of the cell can be calculated by the Nernst equation:

$ E = E^{\circ} - \frac{RT}{nF} \ln\left(\frac{[M^{2+}]_{\text{cathode}}}{[M^{2+}]_{\text{anode}}}\right) $

Since it's a concentration cell, $ E^\circ = 0 $. At 298 K, substituting from the provided conversion (2.303 $ \times $ R $ \times $ 298/F = 0.059 V) and given n = 2 (because two electrons are transferred per metal ion):

$ E = - \frac{(0.059)}{2} \log\left(\frac{[M^{2+}]_{saturated}}{[M^{2+}]_{0.001 \text{ M}}}\right) $

Given $ E = 0.059 $ V, we can solve for the concentration of $ M^{2+} $ in the saturated solution:

$ 0.059 = -0.0295 \log\left(\frac{[M^{2+}]_{saturated}}{0.001}\right) $

$ \log\left(\frac{[M^{2+}]_{saturated}}{0.001}\right) = -2 $

$ [M^{2+}]_{saturated} = 0.001 \times 10^{-2} $

$ [M^{2+}]_{saturated} = 0.00001 \text{ M} $

The solubility product, K_sp, of MX₂ can now be computed. Let s be the solubility of MX₂ in mol/L. The dissolution of MX₂ is given by:

$ MX_2 \rightleftharpoons M^{2+} + 2 X^- $

$ \quad s \quad\quad\quad s \quad \quad \, 2s $

The solubility product expression is:

$ K_{sp} = [M^{2+}][X^-]^2 = (s)(2s)^2 $

$ K_{sp} = 4s^3 $

Since $ s = [M^{2+}]_{saturated} = 0.00001 $ M,

$ K_{sp} = 4(0.00001)^3 $

$ K_{sp} = 4 \times 10^{-15} $

Thus, the solubility product of MX₂ at 298 K is $ 4 \times 10^{-15} $ mol³ dm^-9, which corresponds to Option B.

The given electrochemical cell is a concentration cell where both the electrodes are of the same metal but immersed in solutions of different concentrations of the same metal ion. The emf (E) generated by this cell can be calculated using the Nernst equation, which for this cell at 298 K (25°C) is given by:

$ E = E^\circ - \frac{0.059}{n} \log \frac{[C_1]}{[C_2]} $

Where:

• $E^\circ$ is the standard electrode potential which is zero in a concentration cell because both electrodes are same.


• $n$ is the number of moles of electrons transferred in the redox reaction (2 in this case, as it involves $M^{2+}$ ions).


• $[C_1]$ and $[C_2]$ are the concentrations of $M^{2+}$ at the two electrodes.


• $E$ is given to be 0.059 V.

Assuming the more concentrated solution is at the left hand electrode and the less concentrated solution (0.001 M) is at the right hand electrode, the Nernst equation simplifies to:

$ E = - \frac{0.059}{2} \log \frac{0.001}{[C_1]} $

To find $[C_1]$, we solve for the argument of log such that the calculated emf matches the given emf (0.059 V):

$ 0.059 = - \frac{0.059}{2} \log \frac{0.001}{[C_1]} $

$ -2 = \log \frac{0.001}{[C_1]} $

$ 10^{-2} = \frac{0.001}{[C_1]} $

$ [C_1] = 0.1 \text{ M} $

This $[C_1]$ value supports the direction of the redox reactions assumed. Now, to find the change in Gibbs free energy $\Delta G$ for this cell, we use the relationship:

$ \Delta G = -nFE $

But $E$ should be positive for the spontaneous reaction, hence:

$ \Delta G = -2 \times 96500 \times 0.059 \, \text{V} = -11381 \, \text{J/mol} = -11.381 \, \text{kJ/mol} $

Therefore, $\Delta G$ for the cell is approximately -11.4 kJ/mol. The correct answer is Option D: -11.4.

The plot obtained from adding AgNO₃(aq.) to a solution of KCl is as follows :

The reaction occurring is :

$ Ag^ + + NO_3^- + K^ + + Cl^- \to AgCl + K^ + + NO_3^- $

Or more succinctly :

$ AgNO_3(aq.) + K^+Cl^-(aq.) \to AgCl(s) + KNO_3 $

Upon the gradual addition of aqueous AgNO₃, precipitation does not begin immediately. The precipitation of AgCl starts only when the ionic product of AgCl exceeds its solubility product. During this initial phase, AgNO₃ precipitates as AgCl, simultaneously adding NO₃^- ions to the solution. Since the total number of ions remains constant, the conductance does not change, represented by the flat segment AB in the figure. Once precipitation is complete, any further addition of AgNO₃ increases the ion concentration in the solution, thus increasing the conductance, shown by the rising segment BC.

To find the cell potential under non-standard conditions, we can use the Nernst equation, which is given as:

$ E = E^o - \frac{RT}{nF} \ln(Q) $

Where:

• $E^o$ is the standard cell potential (1.67 V).
• $R$ is the gas constant (8.314 J/mol·K).
• $T$ is the temperature in Kelvin (298 K for 25°C).
• $n$ is the number of moles of electrons transferred per mole of reaction (4 in this case).
• $F$ is the Faraday constant (96485 C/mol).
• $Q$ is the reaction quotient.

The reaction quotient, $Q$, can be calculated based on the given conditions and the reaction:

$ Q = \frac{[\text{Fe}^{2+}]^2}{[\text{H}^+]^4 \cdot P(O_2)} $

Substituting the given values:

• $[\text{Fe}^{2+}] = 10^{-3} \text{ M}$
• $[\text{H}^+] = 10^{-\text{pH}} = 10^{-3} \text{ M}$
• $P(O_2) = 0.1 \text{ atm}$

$ Q = \frac{(10^{-3})^2}{(10^{-3})^4 \cdot 0.1} $

$ Q = \frac{10^{-6}}{10^{-12} \cdot 0.1} $

$ Q = \frac{10^{-6}}{10^{-13}} $

$ Q = 10^{7} $

Now, substituting the values into the Nernst equation:

$ E = 1.67 \text{ V} - \frac{8.314 \times 298}{4 \times 96485} \ln(10^{7})$

Calculating the term $\frac{RT}{nF}$;

$ \frac{RT}{nF} = \frac{8.314 \times 298}{4 \times 96485} $

$ \approx \frac{2476}{385940} $

$ \approx 0.0064 \text{ V} $

The natural logarithm of $10^{7}$ is about 16.1 (since $\ln(10) \approx 2.303$, so $\ln(10^{7}) = 7 \times \ln(10) \approx 16.1$).

Thus,

$ E = 1.67 \text{ V} - 0.0064 \times 16.1$

$ E = 1.67 \text{ V} - 0.103 \text{ V} $

$ E = 1.567 \text{ V} $

Therefore, the cell potential under the given conditions is approximately 1.57 V, so option D (1.57 V) is the correct answer.

To understand how the concentration of ions affects the cell potential, we will first use the Nernst equation. The Nernst equation gives us a way to calculate the potential of a cell under non-standard conditions and is represented as follows:

$ E = E^0 - \frac{RT}{nF} \ln \frac{a_{\text{Red}}}{a_{\text{Ox}}} $

In our case, we're analyzing a concentration cell where the metal M is the same in both the anode and the cathode but with different concentrations of M⁺. Here, the standard potential $ E^0 $ is zero because the same substance is used as both the anode and cathode.

The cell reaction becomes:

$ \text{M}(s) | \text{M}^{+}(0.05 \text{ M at anode}) \Longrightarrow \text{M}(s) | \text{M}^{+}(1 \text{ M at cathode}) $

Since $ E^0 = 0 $, the Nernst equation simplifies to:

$ E = - \frac{RT}{nF} \ln \frac{[\text{M}^{+} \text{ (cathode)}]}{[\text{M}^{+} \text{ (anode)}]} $

Which turns into:

$ E = - \frac{RT}{nF} \ln \frac{1}{0.05} , $

and we can further simplify using $ \ln(\frac{1}{0.05}) = -\ln(0.05) $. Assume the reaction involves the transfer of 1 mole of electrons (n=1), the value of F (Faraday constant) is approximately 96485 C/mol, and the temperature T is 298K (room temperature). The gas constant R is 8.314 J/(mol·K). Plugging in these values:

$ E \approx - \frac{(8.314 \, \text{J/mol·K})(298 \, \text{K})}{(1)(96485 \, \text{C/mol})} \ln(0.05) $

This equation can be used to find the potential in volts when the concentration at anode was 0.05 M and was resulting in 70 mV.

Now, switching the anode concentration to 0.0025 M, we reapply the Nernst equation:

$ E = - \frac{RT}{nF} \ln \frac{1}{0.0025} $

We can see the ratio of concentrations has changed, moving from 1/0.05 to 1/0.0025. Thus, the concentration difference across the membrane has increased, which should increase the voltage according to the Nernst equation. Specifically, the ratio changed by a factor of 0.05/0.0025 = 20 times.

Given that $\ln(\frac{1}{x})$ is proportional to the voltage, when the ratio of concentration changes by twenty-fold, and because $\ln(0.0025) = -\ln(400)$ which is twice the $\ln(20)$, the potential will double. Since the initial 70 mV doubles, the new cell potential will be:

$ E \approx 2 \times 70 \, \text{mV} = 140 \, \text{mV}. $

Therefore, the correct answer is:

Option C: 140 mV

To determine the right option for the given concentration cell, we must first understand the concepts of cell potential, Gibbs free energy, and the relationship between these quantities.

The cell potential, $ E_{cell} $, for a concentration cell like the one described is directly related to the concentrations of the ions on both sides of the cell. It’s calculated using the Nernst equation:

$ E_{cell} = E^o - \frac{RT}{nF} \ln \frac{[M^+](\text{low concentration})}{[M^+](\text{high concentration})} $

Since the given cell involves the same metal on both sides in its standard state, the standard cell potential, $ E^o $, is zero. Therefore, the equation simplifies to:

$ E_{cell} = - \frac{RT}{nF} \ln \frac{0.05}{1} $

Given that $ R $ (the ideal gas constant) is approximately 8.314 J/mol·K, $ F $ (the Faraday constant) is about 96485 C/mol, $ T $ is the temperature in Kelvin (assuming standard temperature of 298 K), and $ n $ (the number of moles of electrons transferred per mole of reaction) is 1, the equation further simplifies to:

$ E_{cell} = - \frac{8.314 \times 298}{96485} \ln \left(\frac{0.05}{1}\right) $

Calculate the ln term:

$ \ln \left(\frac{0.05}{1}\right) = \ln(0.05) \approx -2.9957 $

Then the equation is:

$ E_{cell} = - \frac{2476.422}{96485} \times -2.9957 \approx 0.077 \text{ volts} = 77 \text{ mV} $

Here, the calculated $ E_{cell} > 0 $, which matches the given cell potential of 70 mV. It differs slightly due to rounding and exact values used in constants. This cell potential being positive indicates spontaneous reaction (tendency to go from high to low concentration).

We now apply this to the relationship between the cell potential and Gibbs free energy, which is given by:

$ \Delta G = -nFE_{cell} $

Since $ E_{cell} > 0 $, $ \Delta G < 0 $ which indicates that the process is thermodynamically favorable (spontaneous).

Looking at the provided options:

• Option A: E_cell < 0 ; $\Delta G > 0$ (Incorrect; as $ E_{cell} $ is positive and $ \Delta G $ is negative)
• Option B: E_cell > 0 ; $\Delta G < 0$ (Correct; matches our calculation)
• Option C: E_cell < 0 ; $\Delta G^o > 0$ (Incorrect)
• Option D: E_cell > 0 ; $\Delta G^o > 0$ (Incorrect, $ \Delta G^o $ not directly relevant here, but centralized on the standard conditions which were zero)

Thus, the correct answer is Option B.

The general criterion for a spontaneous redox reaction is that the reduction potential of the reducing agent (which gets oxidized) must be lower than the reduction potential of the oxidizing agent (which gets reduced). Here, the reduction of NO₃^- has an E⁰ of +0.96 V. For a metal ion to be oxidized by NO₃^-, the E⁰ of the metal ion’s reduction must be less than +0.96 V.

Let's check each given ion to see if it can be oxidized:

• V²⁺ (E⁰ = -1.19 V): This potential is significantly lower than +0.96 V, thus vanadium can be oxidized by nitrate, as it is much easier to reduce NO₃^- than to reduce V²⁺ to V.
• Fe³⁺ (E⁰ = -0.04 V): This potential is also lower than +0.96 V, so iron can be oxidized by nitrate.
• Au³⁺ (E⁰ = +1.40 V): Since the potential for Au³⁺ is higher than the potential for nitrate reduction, gold cannot be oxidized by nitrate. It means nitrate cannot provide sufficient potential to reduce Au³⁺.
• Hg²⁺ (E⁰ = +0.86 V): This potential is close but still lower than +0.96 V, therefore mercury can theoretically be oxidized by nitrate, though it is only slightly easier to reduce NO₃^- than to reduce Hg²⁺.

Given this analysis:

• Option A (V and Hg): Correct, as both V and Hg have lower reduction potentials than the nitrate ion.
• Option B (Hg and Fe): Correct, as analyzed above, both can indeed be oxidized by nitrate.
• Option C (Fe and Au): Incorrect because Au³⁺ has a higher potential and thus cannot be oxidized by nitrate.
• Option D (Fe and V): Correct, given that both V and Fe have lower potentials than nitrate reduction.

The correct answer includes Option A, Option B, and Option D.

Given,

Current passed a 10 mA.

Moles of Hydrogen released are 0.01 moles.

In the electrolysis of aqueous solution sodium chloride, as the reduction potential of sodium ion is less than the hydrogen ion, the reaction that takes place at the cathode could be given as,

2H$^+$ + 2e$^-$ $\to$ H$_2$

0.01 moles of hydrogen would require 2 $\times$ 0.01 = 0.02 moles of electrons (or) 0.02 Faradays.

We know the formula to calculate the amount of Faradays passed.

Amount of Faraday's passed = $\frac{I(A)\times t(s)}{96500~\mathrm{C~mol^{-1}}}$

On rearranging the equation to calculate the time taken we get,

t = Amount of Faraday's passed $\times$ 96500 $\times$ I

$0.02 = {{{{10\,mA} \over {1000\,mA/A}} \times t(s)} \over {96500\,C\,mo{l^{ - 1}}}}$

$\Rightarrow~t = 0.02\times96500\times0.01$

On multiplying we get,

$\Rightarrow~t=19.3\times10^4$ s

The time required to liberate 0.01 moles of H$_2$ gas at the cathode is 19.3 $\times$ 10$^4$ s.

The $\mathrm{E}^{\mathrm{o}}$ value for $\mathrm{I}_{2} / \mathrm{I}^{-}$ half cell is much lower than that for $\mathrm{Cl}_{2} / \mathrm{Cl}^{-}$ half cell. Therefore, $\mathrm{Cl}_{2}$ has a higher tendency to get reduced.

It can therefore oxidise $\mathrm{I}^{-}$ ion to $\mathrm{I}_{2}$.

The cell emf corresponding to this reaction's

$\begin{aligned} \mathrm{E}_{\text {cell }}^{\mathrm{o}} & =\mathrm{E}_{\text {cathode }}^{\mathrm{o}}-\mathrm{E}_{\text {anode }}^{\mathrm{o}}=1.36 \mathrm{~V}-0.54 \mathrm{~V} \\ & =0.82 \mathrm{~V} \end{aligned}$

Positive value of $\mathrm{E}_{\text {cell }}^{\mathrm{o}}$ shows that this reaction is spontaneous in the direction as written.

Stability of $\mathrm{Fe}^{3+}$ and $\mathrm{Mn}^{3+}$ in Acidic Solution

$\mathrm{Fe}^{3+}$ is stable in acid, but $\mathrm{Mn}^{3+}$ is not. Let's see why.

Standard Electrode Potentials

The standard electrode potential, $\mathrm{E}^{\mathrm{o}}$, is a measure of how easily an ion can be reduced. For $\mathrm{Mn}^{3+} \to \mathrm{Mn}^{2+}$, $\mathrm{E}^{\mathrm{o}}$ is a high positive value. This means $\mathrm{Mn}^{3+}$ is eager to gain electrons and become $\mathrm{Mn}^{2+}$.

Comparing Potentials

The half-cell for $\mathrm{O}_2, \mathrm{H}^+/ \mathrm{H}_2 \mathrm{O}$ has a lower $\mathrm{E}^{\mathrm{o}}$ value than for $\mathrm{Mn}^{3+}$. So, $\mathrm{Mn}^{3+}$ can take electrons from water, causing water to split into $\mathrm{O}_2$ and $\mathrm{H}^+$.

The Reaction and Its Cell Potential

Here is the reaction:

$ \begin{aligned} 4 \mathrm{Mn}^{3+} +2 \mathrm{H}_{2} \mathrm{O} \rightarrow 4 \mathrm{Mn}^{2+}+4 \mathrm{H}^{+}+\mathrm{O}_{2} \\ \mathrm{E}_{\text {cell}}^{\mathrm{o}} =\mathrm{E}_{\mathrm{Mn}^{3+} / \mathrm{Mn}}^{\mathrm{o}}-\mathrm{E}_{\mathrm{H}_{2} \mathrm{O}, \mathrm{H}^{+} / \mathrm{O}_{2}}^{\mathrm{o}} \\ =1.50 ~\mathrm{V} - 1.23 ~\mathrm{V} \\ =0.27 ~\mathrm{V} \end{aligned} $

The $\mathrm{E}_{\text{cell}}^{\mathrm{o}}$ is positive (0.27 V), which means the reaction happens easily (is spontaneous).

Conclusion

Since $\mathrm{Mn}^{3+}$ reacts with water and changes into $\mathrm{Mn}^{2+}$, it is not stable in acid. In contrast, $\mathrm{Fe}^{3+}$ does not do this and is stable.

Aniline contains nitrogen in addition to carbon and hydrogen. During fusion with Na metal, the following reaction takes place

$\mathrm{C}+\mathrm{N}+\mathrm{Na} \stackrel{\text { Fuse }}{\longrightarrow} \mathrm{NaCN}$

During the test, following reactions take place.

$6 \mathrm{NaCN}+\mathrm{FeSO}_{4} \rightarrow \mathrm{Na}_{4}\left[\mathrm{Fe}(\mathrm{CN})_{6}\right]+\mathrm{Na}_{2} \mathrm{SO}_{4} \mathrm{Na}_{4}\left[\mathrm{Fe}(\mathrm{CN})_{6}\right]+2 \mathrm{Fe}_{2}\left(\mathrm{SO}_{4}\right)_{3} \rightarrow $ $\mathrm{\mathop {F{e_4}{{[Fe{{(CN)}_6}]}_3}}\limits_{\Pr ussian\,Blue}}$ $+~6 \mathrm{Na}_{2} \mathrm{SO}_{4}$

Moles of NaCl = 4 $\times$ 0.5 = 2 moles (Since 500 ml = 0.5 L)

2 moles of NaCl = 2 moles of Cl$^-$

2Cl$^-$ $\to$ Cl$_2$ + 2e$^-$

This equation clears that 2 moles of Cl$^-$ yields 1 mole of Cl$_2$ gas

Therefore, 1 mole of Cl$_2$ gas is evolved.

Number of moles of Na$^+$ = 2

2NaCl = 2Na

2Na + 2Hg $\to$ 2Na(Hg)

By electrolysis we get 2 moles of sodium which can combine with exactly 2 moles of mercury to give 2 moles of amalgam.

$\therefore$ Maximum weight of Na amalgam (assuming equimolar Na and Hg) = 46 + 400 = 446 g

Na$^+$ + e$^-$ $\to$ Na

Moles of Na discharged at cathode = 2

$\therefore$ The number of electron required for this purpose = 2 moles

$\therefore$ Total charge required = 2 faraday = 2 $\times$ 96500 = 193000 coulombs.

The value of $\mathrm{E}_{\text {cell }}^{\mathrm{o}}$ can also be calculated using following formula.

$ \mathrm{E}_{\text {cell }}^{\mathrm{o}}=\mathrm{E}_{\text {anode }}^{\mathrm{o}}-\mathrm{E}_{\text {cathode }}^{\mathrm{o}} $

The half-reaction at anode is:

$ \begin{aligned} \mathrm{C}_6 \mathrm{H}_{12} \mathrm{O}_6+\mathrm{H}_2 \mathrm{O} & \rightarrow \text { Gluconic acid }\left(\mathrm{C}_6 \mathrm{H}_{12} \mathrm{O}_7\right) \\ & +2 \mathrm{H}^{+}+2 e^{-} ; \mathrm{E}_{\mathrm{oxd}}^{\mathrm{o}}=-0.05 \mathrm{~V} \\ \mathrm{E}_{\text {Red }}^{\mathrm{o}}=\mathrm{E}_{\text {anode }}^{\mathrm{o}} & =0.05 \mathrm{~V} \end{aligned} $

The half-reaction at cathode is:

$ \begin{aligned} & \mathrm{Ag}^{+}+e^{-} \rightarrow \mathrm{Ag} ; \mathrm{E}_{\mathrm{red}}^{\mathrm{o}}=0.8 \mathrm{~V} \\ & \mathrm{E}_{\text {cathode }}^{\mathrm{o}}=+0.8 \mathrm{~V} \end{aligned} $

From the above half-reactions,

$ \begin{aligned} \mathrm{E}_{\text {anode }}^{\mathrm{o}} & =+0.05 \mathrm{~V} \\ \mathrm{E}_{\text {cathode }}^{\mathrm{o}} & =0.8 \mathrm{~V} \end{aligned} $

Substitute the values in above equation for $\mathrm{E}_{\text {cell }}^{\mathrm{o}}$

$ \mathrm{E}_{\text {cell }}^{\mathrm{o}}=(-0.8)-(-0.05) \mathrm{V}=0.75 \mathrm{~V} $

The value of $\ln K$ can be calculated alternatively as follows:

$ \mathrm{E}_{\text {cell }}^{\mathrm{o}}=\frac{\mathrm{RT}}{n \mathrm{~F}} \ln \mathrm{~K} $

As given, $\frac{\mathrm{F}}{\mathrm{RT}}=38.92, n=2$ and $\mathrm{E}_{\text {cell }}^{\mathrm{o}}=0.75 \mathrm{~V}$

Substitute the values in above equation,

$ \begin{aligned} & 0.75=\frac{1}{2 \times 38.92} \ln K \\ & 0.75=\frac{1}{77.84} \ln K \end{aligned} $

On rearranging above expression,

$ \ln K=0.75 \times 77.84=58.38 $

When concentration of $\mathrm{NH}_3$ increases, the concentration of $\mathrm{H}^{+}$decreases.

As per Nernst equation,

$ \mathrm{E}_{\text {red }}=\mathrm{E}_{\text {red }}^{\mathrm{o}}-\frac{0.0592}{n} \log _{10}\left[\mathrm{H}^{+}\right]^2 \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,...(i)$

The $\mathrm{E}_{\text {red }}$ is the single electrode reduction potential and $E_{\text {red }}^o$ is the standard electrode reduction potential. Here $n$ refers to the number of moles of electrons used in the chemical reaction.

Given, $\mathrm{pH}=11$

From pH , the concentration of $\mathrm{H}^{+}$ions can be calculated.

$ \begin{array}{rlrl} \mathrm{pH} & =-\log _{10}\left[\mathrm{H}^{+}\right] \text {and }\left[\mathrm{H}^{+}\right]=10^{-\mathrm{pH}} \\ \therefore \quad & {\left[\mathrm{H}^{+}\right]} =10^{-11} \mathrm{M} \end{array} $

For hydrogen ions, $\mathrm{E}_{\text {oxd }}^{\mathrm{o}}=0$

Substitute the values in equation (i),

$ \begin{array}{ll} & \mathrm{E}_{\text {oxd }}=0-\frac{0.0592}{2} 2 \log _{10} 10^{-11} \\ \therefore & \mathrm{E}_{\text {oxd }}=0.0592 \times 11 \\ \therefore & \mathrm{E}_{\text {oxd }}=0.65 \end{array} $

On increasing the concentration of ammonia, oxidation potential of half cell reaction increases by a factor of +0.65 V .

This shows that, $\mathrm{E}_{\text {red }}$ increase by the factor of 0.65 from $\mathrm{E}_{\text {oxd }}$.

$ \text { The overall chemical equation is: } $

$ \begin{array}{r} 2 \mathrm{Ag}+\mathrm{C}_6 \mathrm{H}_{12} \mathrm{O}_6+\mathrm{H}_2 \mathrm{O} \rightarrow 2 \mathrm{Ag}(s)+\mathrm{C}_6 \mathrm{H}_{12} \mathrm{O}_7+2 \mathrm{H}^{+} \end{array} $

During the above chemical reaction, ammonia is added.

When ammonia reacts with $\mathrm{Ag}^{+}$ions, it produces $\left[\mathrm{Ag}\left(\mathrm{NH}_3\right)_2\right]^{+}$complex. Hence, the information given in option A is correct.

Also, $\left[\mathrm{Ag}\left(\mathrm{NH}_3\right)_2\right]^{+}$is a stronger oxidising reagent as compared to that of $\mathrm{Ag}^{+}$ions. This is because in absence of ammonia (alkaline medium) most of the silver ion precipitates as silver oxide which is not a good oxidizing agent. Hence, option B is correct.

During the above chemical reaction, the glucose is oxidised to gluconic acid as shown below.

$ \begin{array}{r} \mathrm{C}_6 \mathrm{H}_{12} \mathrm{O}_6+\mathrm{H}_2 \mathrm{O} \rightarrow \text { Gluconic acid }\left(\mathrm{C}_6 \mathrm{H}_{12} \mathrm{O}_7\right) +2 \mathrm{H}^{+}+2 e^{-} \end{array} $

If ammonia is not added during this chemical reaction, the silver ions will react with gluconic acid producing the silver salt of gluconic acid. Hence, option C is correct.

The ammonia added during this reaction does not affect standard reduction potential of glucose or gluconic acid electrode.