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Chapter 14: Problem 18

A gas mixture consists of \(8 \mathrm{kmol}\) of \(\mathrm{H}_{2}\) and \(2 \mathrm{kmol}\) of \(\mathrm{N}_{2}\). Determine the mass of each gas and the apparent gas constant of the mixture.

Short Answer

Expert verified
Based on the given mixture of two gases H₂ and N₂ with their amounts in kmol, we have found the mass of H₂ to be 16000 g, the mass of N₂ to be 56000 g, and the apparent gas constant of the mixture to be 3.381 J/(g·K).

Step by step solution

01

Calculate the molecular weight of each gas

To calculate the molecular weight, we use the atomic weight of each element in the gas. Atomic weight of hydrogen, H, is 1 g/mol, and atomic weight of nitrogen, N, is 14 g/mol. Therefore, molecular weight of H₂ is \(2\times1 = 2 \,\mathrm{g/mol}\) and molecular weight of N₂ is \(2\times 14 = 28\, \mathrm{g/mol}\).
02

Calculate the mass of each gas

Using the molecular weights, we will now calculate the mass of each gas in the mixture. Since 1 kmol = 1000 mol, we have: Mass of H₂ = Amount of H₂ (in kmol) × Molecular weight of H₂ × 1000 \( \Rightarrow m_{H2}=8\times2\times1000=16000\, \mathrm{g}\) Mass of N₂ = Amount of N₂ (in kmol) × Molecular weight of N₂ × 1000 \( \Rightarrow m_{N2}=2\times28\times1000=56000\, \mathrm{g}\)
03

Calculate the total mass

The total mass of the mixture is the sum of the mass of each gas. Total mass, \(m_{total}=m_{H2}+m_{N2}=16000\,\mathrm{g} + 56000\,\mathrm{g} = 72000\, \mathrm{g}\)
04

Calculate the mole fraction of each gas

Mole fraction, \(x_i\), of each gas component is defined as the ratio of that gas's moles to the total moles in the mixture. So we have: \(x_{H2}=\frac{8}{8+2} =0.8\) \(x_{N2}=\frac{2}{8+2} =0.2\)
05

Determine the gas constant of each gas

The gas constant for each gas component can be calculated using the molecular weight for that gas and the universal gas constant, \(R_u = 8.314 \, \mathrm{J/(mol\cdot K)}\). \(R_{H2} =\frac{R_u}{M_{H2}}=\frac{8.314\, \mathrm{J/(mol\cdot K)}}{2\, \mathrm{g/mol}}=4.157\, \mathrm{J/(g\cdot K)}\) \(R_{N2} =\frac{R_u}{M_{N2}}=\frac{8.314\, \mathrm{J/(mol\cdot K)}}{28\, \mathrm{g/mol}}=0.297\, \mathrm{J/(g\cdot K)}\)
06

Calculate the apparent gas constant of the mixture

The apparent gas constant, \(R_m\), of the mixture is given by the sum of the product of mole fractions and gas constants for each gas component in the mixture. \(R_m=x_{H2}\times R_{H2} + x_{N2}\times R_{N2}=0.8 \times 4.157\, \mathrm{J/(g\cdot K)} + 0.2 \times 0.297\, \mathrm{J/(g\cdot K)}=3.381\, \mathrm{J/(g\cdot K)}\) The mass of H₂ is \(16000\, \mathrm{g}\), the mass of N₂ is \(56000\, \mathrm{g}\), and the apparent gas constant of the mixture is \(3.381\, \mathrm{J/(g\cdot K)}\).

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Most popular questions from this chapter

A recent attempt to circumnavigate the world in a balloon used a helium-filled balloon whose volume was \(7240 \mathrm{~m}^{3}\) and surface area was \(1800 \mathrm{~m}^{2}\). The skin of this balloon is \(2 \mathrm{~mm}\) thick and is made of a material whose helium diffusion coefficient is \(1 \times 10^{-9} \mathrm{~m}^{2} / \mathrm{s}\). The molar concentration of the helium at the inner surface of the balloon skin is \(0.2 \mathrm{kmol} / \mathrm{m}^{3}\) and the molar concentration at the outer surface is extremely small. The rate at which helium is lost from this balloon is (a) \(0.26 \mathrm{~kg} / \mathrm{h}\) (b) \(1.5 \mathrm{~kg} / \mathrm{h}\) (c) \(2.6 \mathrm{~kg} / \mathrm{h}\) (d) \(3.8 \mathrm{~kg} / \mathrm{h}\) (e) \(5.2 \mathrm{~kg} / \mathrm{h}\)

Using Henry's law, show that the dissolved gases in a liquid can be driven off by heating the liquid.

Heat convection is expressed by Newton's law of cooling as \(\dot{Q}=h A_{s}\left(T_{s}-T_{\infty}\right)\). Express mass convection in an analogous manner on a mass basis, and identify all the quantities in the expression and state their units.

14-45 Consider a rubber membrane separating carbon dioxide gas that is maintained on one side at \(2 \mathrm{~atm}\) and on the opposite at \(1 \mathrm{~atm}\). If the temperature is constant at \(25^{\circ} \mathrm{C}\), determine (a) the molar densities of carbon dioxide in the rubber membrane on both sides and \((b)\) the molar densities of carbon dioxide outside the rubber membrane on both sides.

When the density of a species \(A\) in a semi-infinite medium is known at the beginning and at the surface, explain how you would determine the concentration of the species \(A\) at a specified location and time.
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