An electrochemical cell consists of a nickel metal electrode...

An electrochemical cell consists of a nickel metal electrode immersed in a solution with $\left[\mathrm{Ni}^{2+}\right]=1.0 M$ separated by a porous disk from an aluminum metal electrode immersed in a solution with $\left[\mathrm{Al}^{3+}\right]=1.0 M .$ Sodium hydroxide is added to the aluminum compartment, causing $\mathrm{Al}(\mathrm{OH})_{3}(s)$ to precipitate. After precipitation of $\mathrm{Al}(\mathrm{OH})_{3}$ has ceased, the concentration of $\mathrm{OH}^{-}$ is $1.0 \times 10^{-4} M$ and the measured cell potential is $1.82 \mathrm{~V}$. Calculate the $K_{\mathrm{sp}}$ value for $\mathrm{Al}(\mathrm{OH})_{3}$. $$\mathrm{Al}(\mathrm{OH})_{3}(s) \rightleftharpoons \mathrm{Al}^{3+}(a q)+3 \mathrm{OH}^{-}(a q) \quad K_{\mathrm{sp}}=?$$

An electrochemical cell consists of a nickel metal electrode immersed in a solution with $\left[\mathrm{Ni}^{2+}\right]=1.0 M$ separated by a porous disk from an aluminum metal electrode immersed in a solution with $\left[\mathrm{Al}^{3+}\right]=1.0 M .$ Sodium hydroxide is added to the aluminum compartment, causing $\mathrm{Al}(\mathrm{OH})_{3}(s)$ to precipitate. After precipitation of $\mathrm{Al}(\mathrm{OH})_{3}$ has ceased, the concentration of $\mathrm{OH}^{-}$ is $1.0 \times 10^{-4} M$ and the measured cell potential is $1.82 \mathrm{~V}$. Calculate the $K_{\mathrm{sp}}$ value for $\mathrm{Al}(\mathrm{OH})_{3}$.
$$\mathrm{Al}(\mathrm{OH})_{3}(s) \rightleftharpoons \mathrm{Al}^{3+}(a q)+3 \mathrm{OH}^{-}(a q) \quad K_{\mathrm{sp}}=?$$

Key Concepts

Solubility Product (Ksp)

The solubility product constant, Ksp, is the equilibrium constant for the dissolution of a sparingly soluble compound. It quantifies the extent to which the compound dissolves in water by relating the concentrations of its constituent ions at equilibrium. Determining the Ksp is essential for predicting the formation of precipitates and understanding the chemical behavior of compounds in various solution conditions.

Precipitation Equilibrium

Precipitation equilibrium refers to the state achieved when a sparingly soluble salt is in balance with its ions in solution. At this point, the rate of dissolution equals the rate of precipitation, and the concentrations of the ions remain constant. This concept is key to controlling the solubility conditions in a reaction system and understanding how the addition of a common ion, such as hydroxide in this case, drives the formation of a precipitate.

Nernst Equation

The Nernst equation provides a relationship between the cell potential and the concentrations (or activities) of the reacting species. It allows one to calculate the cell’s voltage under non?standard conditions by accounting for the effect of reactant and product concentrations on the redox potentials. This equation is crucial in electrochemistry for determining equilibrium constants and ion concentrations from measured potentials.

Electrochemical Cell

An electrochemical cell is a system where chemical reactions produce electrical energy or vice versa. It consists of two electrodes connected by an electrolyte and often separated by a porous barrier, which facilitates ionic conduction while preventing direct contact between different chemical environments. This concept is fundamental for linking chemical equilibria and redox reactions to measurable electric potentials.

Transcript

00:01 So for this one, we're trying to figure out what ksp is for this aluminum compound.

00:08 And so first, you want to write out the expression for calculating ksp, which is going to be aluminum concentration times hydroxide cubed.

00:16 So we are given the hydroxide concentration.

00:19 We need the aluminum concentration.

00:21 But the thing is we can't just assume the aluminum concentration is just going to be hydroxide divided by three.

00:29 We need to figure out what it's going to be because this is an electrochemical equation.

00:34 It's not just dissolving in solution.

00:37 It's this whole electrochemical cell has been set up.

00:41 So we need to use the nerx equation to figure out the aluminum concentration.

00:46 And so the aluminum concentration is actually embedded within q.

00:52 So we have to get, we have to isolate that q variable and then from there we can get aluminum concentration.

01:02 So this is where we are given stuff from the problem.

01:08 So this is going to be e is 1 .82 volts.

01:13 E0 is going to be 1 .38 volts.

01:16 And then the moles of electrons is 6, and that is going to be 0 .0591 divided by 6 is 0 .009 .5 .5 .5.

01:31 And then now we just kind of leave this q part alone for now.

01:36 We could start moving things over with algebra, so we can subtract the 1 .38, and then we can divide by the negative 0 .00985.

01:50 So that gives us this purple part here.

01:53 Negative 44 .7 is equal to the log of q.

01:56 We can see where that aluminum concentration is hiding.

01:59 We need to get rid of the log, and we can do that...

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