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Chapter 14: Problem 42

Hydrogen gas at \(750 \mathrm{kPa}\) and \(85^{\circ} \mathrm{C}\) is stored in a spherical nickel vessel. The vessel is situated in a surrounding of atmospheric air at \(1 \mathrm{~atm}\). Determine the molar and mass concentrations of hydrogen in the nickel at the inner and outer surfaces of the vessel.

Short Answer

Expert verified
Answer: According to Siegert's law, the solubility of hydrogen in a metal is directly proportional to the square root of the pressure of the surrounding hydrogen gas.

Step by step solution

01

Identifying the given information

We know the following: - Pressure of hydrogen gas inside the vessel: \(P_{H2} = 750 \mathrm{kPa}\) - Temperature of hydrogen gas inside the vessel: \(T = 85^{\circ} \mathrm{C}\) - Pressure of atmospheric air outside the vessel: \(P_{air} = 1 \mathrm{~atm}\) Converting the given temperature to Kelvin: \(T(K) = T(^\circ C) + 273.15\)
02

Convert the pressures to the same unit

Convert \(P_{air}\) to kPa, since \(P_{H2}\) is given in kPa. \(P_{air} = 1 \mathrm{~atm} \times \frac{101.3 \mathrm{kPa}}{1 \mathrm{~atm}} = 101.3 \mathrm{kPa}\) Now we have both pressures in kPa: - \(P_{H2} = 750 \mathrm{kPa}\) - \(P_{air} = 101.3 \mathrm{kPa}\)
03

Calculate the solubility of hydrogen in the nickel (Siegert's law)

Siegert's law states that the solubility of hydrogen in a metal is directly proportional to the square root of the pressure of the surrounding hydrogen gas. Mathematically, it is represented as: \(C = k \sqrt{P}\) where \(C\) is the concentration of hydrogen in the metal, \(k\) is a proportionality constant, and \(P\) is the pressure of the surrounding hydrogen gas. We will need to apply this equation at the inner and outer surfaces of the vessel.
04

Calculate the molar concentration of hydrogen at the inner and outer surfaces

Using Siegert's law, we can find the molar concentration of hydrogen at the inner and outer surfaces: At the inner surface: \(C_{inner} = k \sqrt{P_{H2}}\) At the outer surface: \(C_{outer} = k \sqrt{P_{air}}\) Note: The proportionality constant, \(k\), will be the same for both the inner and the outer surfaces.
05

Calculate the mass concentration of hydrogen at the inner and outer surfaces

To find the mass concentration of hydrogen, we need to multiply the molar concentration by the molar mass of hydrogen (\(M_{H2} = 2.016 \mathrm{g/mol}\)): At the inner surface: \(mass\_conc_{inner} = C_{inner} \times M_{H2} = k \sqrt{P_{H2}} \times M_{H2}\) At the outer surface: \(mass\_conc_{outer} = C_{outer} \times M_{H2} = k \sqrt{P_{air}} \times M_{H2}\) In conclusion, to determine the molar and mass concentrations of hydrogen at the inner and outer surfaces of the vessel, we need to know the proportionality constant \(k\). Once we have the value of \(k\), we can use Siegert's law and the given pressures to calculate the concentrations.

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