Question

For the following half-reaction, \(\mathscr{E}^{\circ}=-2.07 \mathrm{~V}\) : $$\mathrm{AlF}_{6}^{3-}+3 \mathrm{e}^{-} \longrightarrow \mathrm{Al}+6 \mathrm{~F}^{-}$$ Using data from Table \(18.1\), calculate the equilibrium constant at \(25^{\circ} \mathrm{C}\) for the reaction $$\mathrm{Al}^{3+}(a q)+6 \mathrm{~F}^{-}(a q) \rightleftharpoons \mathrm{AlF}_{6}^{3-}(a q) \quad K=?$$

Short answer

The equilibrium constant, \( K \), at \( 25^\circ\mathrm{C} \) for the reaction $$\mathrm{Al}^{3+}(a q)+6 \mathrm{~F}^{-}(a q) \rightleftharpoons \mathrm{AlF}_{6}^{3-}(a q)$$ is approximately \( 1.16 \times 10^{24} \).

Step-by-step solution

Write down the Nernst Equation

The Nernst Equation relates the standard cell potential, \(\mathscr{E}^\circ\), to the Gibbs free energy change, \(\Delta G^\circ\). It can be expressed as follows: $$ \Delta G^\circ = -nFE^\circ $$ where, - n is the number of moles of electrons involved in the reaction - F is the Faraday constant, which equals \( 96,485 \frac{J}{V \cdot mol} \) - \(\mathscr{E}^\circ\) is the standard cell potential

Calculate the number of moles of electrons involved in the redox reaction

From the half-reaction, we can see that 3 moles of electrons are involved in the process: $$ \mathrm{AlF}_{6}^{3-} + 3\mathrm{e}^- \longrightarrow \mathrm{Al} + 6\mathrm{F}^- $$ So, n = 3

Calculate the standard Gibbs free energy change

Now, we can use the Nernst Equation to calculate the standard Gibbs free energy change using the standard cell potential given: $$ \Delta G^\circ = -nFE^\circ $$ $$ \Delta G^\circ = -3 \times 96485 \times (-2.07\,\mathrm{V}) $$ Calculating this yields: $$ \Delta G^\circ = 601,104.05\, \mathrm{J/mol} $$

Calculate the equilibrium constant

We can now use the standard Gibbs free energy change to find the equilibrium constant using the formula: $$ K = e^{\left(-\frac{\Delta G^{\circ}}{RT}\right)} $$ where, - R is the gas constant, which equals \( 8.314 \mathrm{J / mol \cdot K} \) - T is the temperature in Kelvin. Given that the temperature is \( 25^\circ\mathrm{C} \), we need to convert it to Kelvin: $$ T = 25 + 273.15 = 298.15\,\mathrm{K} $$ Now, we can calculate the equilibrium constant: $$ K = e^{\left(-\frac{601,104.05\, \mathrm{J/mol}}{8.314\, \mathrm{J/mol\cdot K} \cdot 298.15\,\mathrm{K}}\right)} $$ Calculating this yields: $$ K \approx 1.16 \times 10^{24} $$ Thus, the equilibrium constant, \( K \), at \( 25^\circ\mathrm{C} \) for the given reaction is approximately \( 1.16 \times 10^{24} \).

Key concepts

Nernst Equation

Understanding the Nernst Equation is crucial for students diving into the world of electrochemistry. Put simply, it's a relationship that connects the concentrations of reactants and products in a chemical reaction to its electrochemical cell potential. What does this mean? Well, imagine you have a battery, and the Nernst Equation can tell you how the voltage changes as the battery is used and the concentration of chemicals inside it shift.

Here's the general form of the Nernst Equation for a reaction at temperature T:
$$ E = E^\circ - \frac{RT}{nF}ln(Q) $$
In this equation, E represents the cell potential at non-standard conditions, while \( E^\circ \) is the standard cell potential. The term R stands for the gas constant, T for the temperature in Kelvin, n for the number of moles of electrons transferred, F is the Faraday constant, and Q is the reaction quotient -- think of it like a snapshot of where the reaction is at any point in time.

By understanding and applying the Nernst Equation, you can predict how the voltage of a cell changes over time, which has profound implications for everything from batteries in electric vehicles to the biological cells in your body.

Gibbs Free Energy

The concept of Gibbs Free Energy is integral in predicting the direction of chemical reactions. It's like a signpost that says whether a reaction will happen spontaneously or if it needs a little push. Represented as \( \Delta G \), it’s a measure of the maximum reversible work a thermodynamic system can perform. It tells us about the balance between enthalpy (the heat energy change), entropy (the disorder within the system), and temperature.

The rule of thumb is simple: If \( \Delta G \) is negative, the reaction is spontaneous in the direction written. If it's positive, you'll need to spend energy to make it happen. This is why knowing the Gibbs Free Energy change is so crucial when considering reactions, including those in electrochemical cells. If we know the standard cell potential and the number of electrons involved, we can use the equation \( \Delta G^\circ = -nFE^\circ \) to calculate \( \Delta G \).

Standard Cell Potential

When we talk about the standard cell potential, \( \mathscr{E}^\circ \), we're looking at the voltage or electromotive force of a galvanic cell under standard conditions. It's the driving force behind the flow of electrons in an electrochemical cell, similar to the pressure pushing water through pipes.

To find \( \mathscr{E}^\circ \), we compare the potential of an electrode to a standard hydrogen electrode which is arbitrarily given a potential of zero volts. The greater the standard cell potential, the greater the cell’s ability to drive an electric current through an external circuit.

This potential is critical in calculating the equilibrium constant for reactions, as it's directly related to the Gibbs free energy change. These relationships are fundamental in understanding how batteries function and how to harness chemical reactions to do work.

Redox Reaction

Lastly, let's demystify the redox reaction. It's a dance between two molecules involving the exchange of electrons — one gives up electrons (oxidation) while the other takes them (reduction). Just like a tango requires two, redox requires two substances to engage in this elegant electron swap.

Each part of a redox reaction can be written as a separate half-reaction, showing either oxidation or reduction. These reactions are not just academic concepts but are at the core of processes like combustion, respiration, and the functioning of batteries. In our exercise scenario, aluminium ions gain electrons (are reduced), while electron donors (redox partners) must be present to maintain balance. Recognizing and balancing these half-reactions is essential for calculating the Nernst Equation and ultimately determining the equilibrium constant for a reaction.