Unit 1 Gases
1.4 Exercises
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Section 1.4 Exercises
- A balloon filled with helium gas is found to take 6 hours to deflate to 50% of its original volume. How long will it take for an identical balloon filled with the same volume of hydrogen gas (instead of helium) to decrease its volume by 50%?
- Starting with the definition of rate of effusion and Graham’s finding relating rate and molar mass, show how to derive the Graham’s law equation, relating the relative rates of effusion for two gases to their molecular masses.
- Which of the following gases diffuse more slowly than oxygen? F2, Ne, N2O, C2H2, NO, Cl2, H2S
- Helium has two naturally isotopes: He-3 and He-4. The atomic mass of He-3 is about 3.00 g/mol and that of He-4 is about 4.00 g/mol. A mixture containing an equal number of moles of He-3 and He-4 is allowed to effuse down a capillary. What percent of the molecules effusing first would be He-4?
- When two cotton plugs, one moistened with ammonia and the other with hydrochloric acid, are simultaneously inserted into opposite ends of a glass tube that is 87.0 cm long, a white ring of NH4Cl forms where gaseous NH3 and gaseous HCl first come into contact. (Hint: Calculate the rates of diffusion for both NH3 and HCl, and find out how much faster NH3 diffuses than HCl.)[latex]\text{NH}_3(g) + \text{HCl}(g) \longrightarrow \text{NH}_4 \text{Cl}(s)[/latex]
At approximately what distance from the ammonia moistened plug does this occur?
- Answer the following questions:
(a) At a pressure of 1 atm and a temperature of 20 °C, dry air has a density of 1.2256 g/L. What is the (average) molar mass of dry air?
(b) The average temperature of the gas in a hot air balloon is 54.4 °C. Calculate its density, assuming the molar mass equals that of dry air.
(c) The lifting capacity of a hot air balloon is equal to the difference in the mass of the cool air displaced by the balloon and the mass of the gas in the balloon. What is the difference in the mass of 1.00 L of the cool air in part (a) and the hot air in part (b)?
(d) An average balloon has a diameter of 18 m and a volume of 3115 m3. What is the lifting power of such a balloon? If the weight of the balloon and its rigging is 227 kg, what is its capacity for carrying passengers and cargo? The volume V of a sphere with radius r is V = (4/3)πr3
Solutions
- 4.2 hours
- Effusion can be defined as the process by which a gas escapes through a pinhole into a vacuum. Graham’s law states that with a mixture of two gases A and B: [latex](\frac{\text{rate A}}{\text{rate B}}) = (\frac{\text{molar mass of B}}{\text{molar mass of A}})^{1/2}[/latex]. Both A and B are in the same container at the same temperature, and therefore will have the same kinetic energy:
[latex]E_{\text{K, A}} = E_{\text{K, B}} \\ E_{\text{K}} = \frac{1}{2} \;mv^2[/latex]
Therefore, [latex]\frac{1}{2}m_{\text{A}}{v^2}_{\text{A}} = \frac{1}{2} m_{\text{B}} {v^2}_{\text{B}}[/latex]
[latex]\begin{array}{r @{{}={}} l} \frac{v^2_\text{A}}{v^2_\text{B}} &= \frac{m_{\text{B}}}{m_{\text{A}}} \\[1em] (\frac{v^2_\text{A}}{v^2_\text{B}})^{1/2} &= (\frac{m_{\text{B}}}{m_{\text{A}}})^{1/2} \\[1em] (\frac{v_\text{A}}{v_\text{B}}) &= (\frac{m_{\text{B}}}{m_{\text{A}}})^{1/2} \end{array}[/latex] - F2, N2O, Cl2, H2S
- The relative rate of effusion is
[latex]\dfrac{\text{rate}_{\text{ He-3}}}{\text{rate}_{\text{ He-4}}}=\sqrt{\dfrac{\text{molar mass}_{\text{ He-4}}}{\text{molar mass}_{\text{ He-3}}}} = \sqrt{\dfrac{4.00 \text{ g/mol}}{3.00 \text{ g/mol}}} = 1.15[/latex]
There is 1.15x amount of He-3 effusing for each x amount of He-4 effusing. The total amount of effusing gases is 1.15x + x. The percent of He-4
[latex]\dfrac{x}{1.15x+x}=0.464=46.4\%[/latex] - 51.7 cm
- (a) 29.48 g mol−1
(b) 1.0966 g L−1
(c) 0.129 g/L
(d) 4.01 × 105 g; net lifting capacity = 174 kg
