Question

Consider the half-reaction ${\mathrm{Ag}}^{+}(aq)+{\mathrm{e}}^{-}⟶\mathrm{Ag}(s)$. Which of the lines in the following diagram indicates how the reduction potential varies with the concentration of ${\mathrm{Ag}}^{+}?(\mathrm{b})$ What is the value of $E_{\mathrm{red}}$ when $\log \left[{\mathrm{Ag}}^{+}\right]=0?$ [Section $20.6]$

Short answer

In the given half-reaction ${\mathrm{Ag}}^{+}(aq)+{\mathrm{e}}^{-}⟶\mathrm{Ag}(s)$, the reduction potential varies with the concentration of ${\mathrm{Ag}}^{+}$ according to the Nernst Equation: $E=E°-\frac{2.303RT}{F}\log [{\mathrm{Ag}}^{+}]$. When $\log [{\mathrm{Ag}}^{+}]=0$, implying the concentration of Ag+ [Ag+] = 1, the reduction potential E_red is equal to the standard reduction potential E° for the given half-reaction.

Step-by-step solution

Understanding the Nernst Equation

The Nernst Equation provides the relationship between the standard reduction potential (E°) and the actual reduction potential (E) for a half-reaction under non-standard conditions. This equation is: $$E=E°-\frac{RT}{nF}\ln Q$$ Where: - E° is the standard reduction potential - R is the gas constant (8.314 J/(mol·K)) - T is the temperature of the reaction (in Kelvin) - n is the number of moles of electrons exchanged in the half-reaction - F is Faraday's constant (96485 C/mol) - Q is the reaction quotient (concentration of products to reactants) For our case, the half-reaction is given by: $${\mathrm{Ag}}^{+}(aq)+{\mathrm{e}}^{-}⟶\mathrm{Ag}(s)$$ Here, n = 1 (i.e., only one electron is exchanged in the half-reaction), and the reaction quotient, Q, is simply equal to the concentration of Ag+ ions, [Ag+].

Updating the Nernst Equation for the Problem

We can rewrite the Nernst Equation for our specific half-reaction as: $$E=E°-\frac{RT}{F}\ln [{\mathrm{Ag}}^{+}]$$ As we are concerned with log[Ag+], we can update the equation by applying the property of logarithm as follows: $$E=E°-\frac{2.303RT}{F}\log [{\mathrm{Ag}}^{+}]$$ Now, we are given $\log [{\mathrm{Ag}}^{+}]=0$, which means that the concentration of Ag+ [Ag+] = 1 (as the base 10 logarithm of 1 equals 0).

Plug in $\log [{\mathrm{Ag}}^{+}]=0$

Substitute the given log[Ag+] value into the equation: $$E=E°-\frac{2.303RT}{F}\cdot 0$$

Simplify and Solve for E_red

Since anything multiplied by 0 equals 0, the equation becomes: $$E=E°$$ In this case, when $\log [{\mathrm{Ag}}^{+}]=0$, the actual reduction potential (E_red;under non-standard condition) is equal to the standard reduction potential (E°) for the given half-reaction.

Key concepts

Reduction Potential

Reduction potential is a fundamental concept when examining redox reactions, such as the conversion of silver ions to solid silver. It measures the tendency of a chemical species to acquire electrons and thereby be reduced. If you think of electrons as 'gifts', the reduction potential tells us how much a species 'wants' to receive these gifts. When dealing with half-reactions, like the one with silver ions, this potential is often represented as $E$.
In redox chemistry, the more positive a reduction potential is, the greater the species' affinity for electrons. Conversely, a more negative value means a lesser tendency to accept electrons. This tells us how spontaneous a reaction will be when connected with other redox pairs.
For the silver ion reaction in particular, a high reduction potential implies silver ions are readily reduced to metallic silver, underlining the reaction’s feasibility under appropriate conditions.

Standard Reduction Potential

Standard reduction potential, denoted as $E^0$, is a crucial reference that chemists use to determine how likely a redox reaction will occur under standard conditions (which are 1 M concentration, 1 atm pressure, and usually 25°C for temperature). This value is a constant and is determined from experimental data for each half-reaction.
In relation to the Nernst equation, $E^0$ serves as the baseline from which the actual reduction potential $E$ is calculated when conditions deviate from standard. For instance, if the concentration of ions changes, the real-time potential can be quite different from the standard value.
By having a value for $E^0$, one can readily calculate the potential for any cell, provided the conditions are known. This helps predict cell voltage in electrochemical circuits.

Reaction Quotient

The reaction quotient, $Q$, is an essential part of understanding the Nernst equation and its application to electrochemistry. Defined as the ratio of the concentrations of the products to the reactants at any point in a reaction, $Q$ provides insight into the current state of the reaction.

• In the specific scenario of the half-reaction involving silver ions, $Q$ would typically be represented by $[{\mathrm{Ag}}^{+}]$, since it's the reversible reactant in this case.
• When $Q$ is calculated for this reaction, it tells us how far the reaction has moved towards equilibrium. A $Q$ value of 1 indicates that the conditions mirror those of standard states.
• Importantly, $Q$ is pivotal in the Nernst equation. Changes in $Q$ reflect direct effects on the potential of a reaction, showing how real-time conditions like concentration shifts can impact the cell potential.

Understanding $Q$ is crucial for predicting how chemical systems behave dynamically and how alterations in conditions alter the path towards equilibrium.