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Chapter 18: Problem 128

Given the following standard reduction potentials, calculate the ion-product, \(K_{w},\) for water at \(25^{\circ} \mathrm{C}\) : $$ \begin{array}{lr} 2 \mathrm{H}^{+}(a q)+2 e^{-} \longrightarrow \mathrm{H}_{2}(g) & E^{\circ}=0.00 \mathrm{~V} \\ 2 \mathrm{H}_{2} \mathrm{O}(l)+2 e^{-} \longrightarrow \mathrm{H}_{2}(g)+2 \mathrm{OH}^{-}(a q) & \\ E^{\circ}=-0.83 \mathrm{~V} \end{array} $$

Short Answer

Expert verified
The ion-product of water at \(25^{\circ} \mathrm{C}\) is calculated by using the above steps, initially deriving the equations for the autoionization of water and then, using the Nernst equation and the provided information, calculating \[K_w\]. Finally, the value is obtained by substitution and evaluation of these parameters in the formula derived.

Step by step solution

01

Derive the equation for autoionization of water

The equation for the autoionization of water can be obtained by subtracting the first equation from the second. This will give us the equation for autoionization of water: \[2H_{2}O (l) \rightarrow H_{2} (g) + 2H^+ (aq) + 2OH^-(aq) \] The standard cell potential (E°) for the auto ionization of water can be calculated by subtracting the E° value of the first equation from that of the second equation. So E° = -0.83V - 0.00V = -0.83V.
02

Calculate the equilibrium constant

The equilibrium constant for this reaction can be expressed as \[K = [H^+][OH^-]\]. Now it's important to remember the Nernst equation, which is given as: \[E° = -(RT/nF) * lnK\] Here, R is the gas constant = 8.314 J/Kmol, T is the temperature = 298 K (25°C is used, which should be converted to Kelvin by adding 273), n is the number of moles of electrons = 2, and F is Faraday's constant = 96500 C/mol. Rewriting the Nernst equation for K, we get: \[K = e^{(-nFE°/RT)}\]
03

Calculate the ion-product of water

Substitute the values for n, R, T, F and E° in the above equation and calculate for K. The K calculated here represents the ion product of water (\[K_w\]) in molar units and thus will allow us to calculate the numerical value of \(K_w\) at the given temperature.

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Most popular questions from this chapter

Predict whether the following reactions would occur spontaneously in aqueous solution at \(25^{\circ} \mathrm{C}\). Assume that the initial concentrations of dissolved species are all \(1.0 M\) (a) \(\mathrm{Ca}(s)+\mathrm{Cd}^{2+}(a q) \longrightarrow \mathrm{Ca}^{2+}(a q)+\mathrm{Cd}(s)\) (b) \(2 \mathrm{Br}^{-}(a q)+\mathrm{Sn}^{2+}(a q) \longrightarrow \mathrm{Br}_{2}(l)+\operatorname{Sn}(s)\) (c) \(2 \mathrm{Ag}(s)+\mathrm{Ni}^{2+}(a q) \longrightarrow 2 \mathrm{Ag}^{+}(a q)+\mathrm{Ni}(s)\) (d) \(\mathrm{Cu}^{+}(a q)+\mathrm{Fe}^{3+}(a q) \longrightarrow\) $$\mathrm{Cu}^{2+}(a q)+\mathrm{Fe}^{2+}(a q)$$

Calcium oxalate \(\left(\mathrm{CaC}_{2} \mathrm{O}_{4}\right)\) is insoluble in water. This property has been used to determine the amount of \(\mathrm{Ca}^{2+}\) ions in blood. The calcium oxalate isolated from blood is dissolved in acid and titrated against a standardized \(\mathrm{KMnO}_{4}\) solution as described in Problem \(18.72 .\) In one test, it is found that the calcium oxalate isolated from a 10.0 -mL sample of blood requires \(24.2 \mathrm{~mL}\) of \(9.56 \times 10^{-4} \mathrm{M} \mathrm{KMnO}_{4}\) for titration. Calculate the number of milligrams of calcium per milliliter of blood.

The \(E_{\text {cell }}^{\circ}\) for the following cell is \(1.54 \mathrm{~V}\) at \(25^{\circ} \mathrm{C}\) : $$\mathrm{U}(s) \mid \mathrm{U}^{3+}(a q)\left\|\mathrm{Ni}^{2+}(a q)\right\| \mathrm{Ni}(s)$$ Calculate the standard reduction potential for the \(\mathrm{U}^{3+} / \mathrm{U}\) half-cell.

A piece of magnesium ribbon and a copper wire are partially immersed in a \(0.1 M \mathrm{HCl}\) solution in a beaker. The metals are joined externally by another piece of metal wire. Bubbles are seen to evolve at both the \(\mathrm{Mg}\) and \(\mathrm{Cu}\) surfaces. (a) Write equations representing the reactions occurring at the metals. (b) What visual evidence would you seek to show that \(\mathrm{Cu}\) is not oxidized to \(\mathrm{Cu}^{2+} ?\) (c) At some stage, NaOH solution is added to the beaker to neutralize the HCl acid. Upon further addition of \(\mathrm{NaOH}\), a white precipitate forms. What is it?

Gold will not dissolve in either concentrated nitric acid or concentrated hydrochloric acid. However, the metal does dissolve in a mixture of the acids (one part \(\mathrm{HNO}_{3}\) and three parts \(\mathrm{HCl}\) by volume \()\) called aqua regia. (a) Write a balanced equation for this reaction. (Hint: Among the products are \(\mathrm{HAuCl}_{4}\) and \(\mathrm{NO}_{2} .\) ) (b) What is the function of \(\mathrm{HCl} ?\)
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