Zirconium is one of the few metals that retains its structural integrity upon exposure to radiation. The fuel rods in most nuclear reactors therefore are often made of zirconium. Answer the following questions about the redox properties of zirconium based on the half-reaction $$ \mathrm{ZrO}_{2} \cdot \mathrm{H}_{2} \mathrm{O}+\mathrm{H}_{2} \mathrm{O}+4 \mathrm{e}^{-} \longrightarrow \mathrm{Zr}+4 \mathrm{OH}^{-} \quad 8^{\circ}=-2.36 \mathrm{V} $$ a. Is zirconium metal capable of reducing water to form hydrogen gas at standard conditions? b. Write a balanced equation for the reduction of water by zirconium. c. Calculate \(\mathscr{G} \circ, \Delta G^{\circ},\) and \(K\) for the reduction of water by zirconium metal. d. The reduction of water by zirconium occurred during the accidents at Three Mile Island in \(1979 .\) The hydrogen produced was successfully vented and no chemical explosion occurred? If \(1.00 \times 10^{3} \mathrm{kg}\) Zreacts, what mass of \(\mathrm{H}_{2}\) is produced? What volume of \(\mathrm{H}_{2}\) at 1.0 \(\mathrm{atm}\) and \(1000 .^{\circ} \mathrm{C}\) is produced? e. At Chernobyl in \(1986,\) hydrogen was produced by the reaction of superheated steam with the graphite reactor core: $$ \mathrm{C}(s)+\mathrm{H}_{2} \mathrm{O}(g) \longrightarrow \mathrm{CO}(g)+\mathrm{H}_{2}(g) $$ It was not possible to prevent a chemical explosion at Chernobyl. In light of this, do you think it was a correct decision to vent the hydrogen and other radioactive gases into the atmosphere at Three Mile Island? Explain.

Step by step solution

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c. Calculating standard Gibbs free energy change, reaction Gibbs free energy change, and the equilibrium constant

From the given standard reduction potentials, we can calculate the standard electromotive force (EMF) of the reaction: $$ E^{\circ} = -2.36\,V - (-0.83\,V) = -1.53\,V $$ To obtain the standard Gibbs free energy change, we use the equation: $$ \Delta G^{\circ} = -nFE^{\circ} $$ Where n = 2 mol of electrons transferred, F is Faraday's constant (96,485 C/mol), and E^{\circ} = -1.53 V $$ \Delta G^{\circ} = -(2)(96,485\,\frac{\mathrm{C}}{\mathrm{mol}})(-1.53\,\mathrm{V}) = 295,261 \,\frac{\mathrm{J}}{\mathrm{mol}} $$ Now to calculate the equilibrium constant (K) we can use: $$ K = \mathrm{e}^{-\frac{\Delta G^{\circ}}{RT}} $$ Where R is the ideal gas constant (8.314 J/(mol K)) and T is the temperature (298 K) $$ K = \mathrm{e}^{-\frac{295,261\,\frac{\mathrm{J}}{\mathrm{mol}}}{(8.314\,\frac{\mathrm{J}}{\mathrm{mol\,K}})(298\,\mathrm{K})}} = 1.5 \times 10^{-25} $$

02

d. Mass and volume of hydrogen gas produced

To find the mass of hydrogen gas produced from the reaction of 1.00 x 10^3 kg of zirconium, we start by calculating the moles of zirconium: $$ \mathrm{moles\,of\,Zr} = \frac{1.00 \times 10^3 \,\mathrm{kg}}{91.22\,\frac{\mathrm{g}}{\mathrm{mol}}} = 1.10 \times 10^4 \,\mathrm{mol} $$ From the balanced equation, we know that 1 mole of zirconium produces 0.5 moles of hydrogen gas. So we find the moles of hydrogen gas produced: $$ \mathrm{moles\,of\,H_{2}} = (0.5)(1.10 \times 10^4 \,\mathrm{mol}) = 5.50 \times 10^3 \mathrm{mol} $$ Now we can find the mass of hydrogen gas produced: $$ \mathrm{mass\,of\,H_{2}} = (5.50 \times 10^3 \,\mathrm{mol})(2.02\,\frac{\mathrm{g}}{\mathrm{mol}}) = 1.11 \times 10^4\,\mathrm{g} $$ To find the volume of hydrogen gas produced at 1.0 atm and 1000 °C, we can use the ideal gas law: $$ PV = nRT $$ Where P is pressure, V is volume, n is moles, R is the gas constant, and T is temperature. $$ (1.0\,\mathrm{atm})V = (5.50 \times 10^3\,\mathrm{mol})(0.0821\,\mathrm{L\,atm/mol\,K})(1273\,\mathrm{K}) $$ $$ V = 5.78 \times 10^5\,\mathrm{L} $$

03

e. Venting decision at Three Mile Island

At Three Mile Island, although venting the hydrogen and other radioactive gases into the atmosphere would have environmental risks, it prevented a chemical explosion similar to what happened at Chernobyl. In Chernobyl, hydrogen was produced by the reaction of superheated steam with the graphite reactor core, which resulted in a chemical explosion. Venting the hydrogen gas produced at Three Mile Island ensured that a chemical explosion did not occur. Considering the alternatives, it was a correct decision to vent the hydrogen and other radioactive gases into the atmosphere at Three Mile Island to prevent further damage and potential loss of life.

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