Question

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

Answer: The molar and mass concentrations of hydrogen in the nickel at the inner and outer surfaces of the vessel are: Inner surface: - Molar concentration: \(750 \times 10^3~mol~m^{-3}\) - Mass concentration: \(756~kg~m^{-3}\) Outer surface: - Molar concentration: \(101.325 \times 10^3~mol~m^{-3}\) - Mass concentration: \(102.132~kg~m^{-3}\).

Step-by-step solution

Convert pressure and temperature to SI units

First, we need to convert the given pressure and temperature to SI units (Pa and K). Pressure, \(P = 750~kPa = 750 \times 10^3~Pa\) Temperature, \(T = 85^{\circ}C = 85 + 273.15 = 358.15~K\)

Calculate the molar concentration of hydrogen

We will use the ideal gas law to find the molar concentration of hydrogen in the nickel at the inner and outer surfaces of the vessel. The ideal gas law is: \(PV = nRT \Rightarrow n = \frac{PV}{RT}\) Where n is the number of moles of hydrogen, R is the universal gas constant, which is \(8.314~\mathrm{JK^{-1}mol^{-1}}\). For the inner surface of the vessel, \(P = 750 \times 10^3~Pa\) and \(T = 358.15~K\). Plugging the values into the equation, we get: \(n_{inner} = \frac{(750 \times 10^3)(8.314 \times 358.15)}{8.314 \times 358.15} = 750 \times 10^3~mol~m^{-3}\) For the outer surface of the vessel, we need to consider the atmospheric air pressure, which is \(1~atm\). We need to convert this to Pa: \(1~atm = 101.325~kPa = 101.325 \times 10^3~Pa\) Now, we can calculate the molar concentration of hydrogen: \(n_{outer} = \frac{(101.325 \times 10^3)(8.314 \times 358.15)}{8.314 \times 358.15} = 101.325 \times 10^3~mol~m^{-3}\)

Convert molar concentration to mass concentration

To find the mass concentration, we need to multiply the molar concentration by the molar mass of hydrogen. The molar mass of hydrogen, M(H), is \(1.008~g~mol^{-1}\) or \(1.008 \times 10^{-3}~kg~mol^{-1}\). Mass concentration of hydrogen at the inner surface: \(C_{inner} = n_{inner} \times M(H) = 750 \times 10^3 \times 1.008 \times 10^{-3} = 756~kg~m^{-3}\) Mass concentration of hydrogen at the outer surface: \(C_{outer} = n_{outer} \times M(H) = 101.325 \times 10^3 \times 1.008 \times 10^{-3} = 102.132~kg~m^{-3}\) The molar and mass concentrations of hydrogen in the nickel at the inner and outer surfaces of the vessel are: Inner surface: - Molar concentration: \(750 \times 10^3~mol~m^{-3}\) - Mass concentration: \(756~kg~m^{-3}\) Outer surface: - Molar concentration: \(101.325 \times 10^3~mol~m^{-3}\) - Mass concentration: \(102.132~kg~m^{-3}\).