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Chapter 5: Problem 64

A hot-air balloon is filled with air to a volume of \(4.00 \times\) \(10^{3} \mathrm{~m}^{3}\) at 745 torr and \(21^{\circ} \mathrm{C}\). The air in the balloon is then heated to \(62^{\circ} \mathrm{C}\), causing the balloon to expand to a volume of \(4.20 \times 10^{3} \mathrm{~m}^{3}\). What is the ratio of the number of moles of air in the heated balloon to the original number of moles of air in the balloon? (Hint: Openings in the balloon allow air to flow in and out. Thus the pressure in the balloon is always the same as that of the atmosphere.)

Short Answer

Expert verified
The ratio of the number of moles of air in the heated balloon to the original number of moles of air in the balloon is approximately 0.921.

Step by step solution

01

Convert temperatures to Kelvin

To use the ideal gas law, the temperatures should be in Kelvin. Convert the Celsius temperatures to Kelvin by adding 273.15. \(T1_{Kelvin} = T1_{Celsius} + 273.15 = 21 + 273.15 = 294.15~K\) \(T2_{Kelvin} = T2_{Celsius} + 273.15 = 62 + 273.15 = 335.15~K\)
02

Relate volume and temperature using ideal gas law

Since we have constant pressure, let's express the ideal gas law relating the initial and final conditions, taking the ratio n2/n1: \(\frac{n2}{n1} = \frac{P2V2T1}{P1V1T2}\) As P1 = P2, they can be canceled out: \(\frac{n2}{n1} = \frac{V2T1}{V1T2}\)
03

Substitute the given values

Now, substitute the given values for V1, V2, T1, and T2: \(\frac{n2}{n1} = \frac{(4.20\times10^{3}~m^3)(294.15~K)}{(4.00\times10^{3}~m^3)(335.15~K)}\)
04

Calculate the ratio

Perform the calculations: \(\frac{n2}{n1} = \frac{1.236\times10^{6} ~m^3K}{1.342\times10^{6} ~m^3K}\) \(\frac{n2}{n1} ≈ 0.921\) The ratio of the number of moles of air in the heated balloon to the original number of moles of air in the balloon is approximately 0.921.

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Most popular questions from this chapter

Some very effective rocket fuels are composed of lightweight liquids. The fuel composed of dimethylhydrazine \(\left[\left(\mathrm{CH}_{3}\right)_{2} \mathrm{~N}_{2} \mathrm{H}_{2}\right]\) mixed with dinitrogen tetroxide was used to power the Lunar Lander in its missions to the moon. The two components react according to the following equation: \(\left(\mathrm{CH}_{3}\right)_{2} \mathrm{~N}_{2} \mathrm{H}_{2}(l)+2 \mathrm{~N}_{2} \mathrm{O}_{4}(l) \longrightarrow 3 \mathrm{~N}_{2}(g)+4 \mathrm{H}_{2} \mathrm{O}(g)+2 \mathrm{CO}_{2}(g)\) If \(150 \mathrm{~g}\) dimethylhydrazine reacts with excess dinitrogen tetroxide and the product gases are collected at \(127^{\circ} \mathrm{C}\) in an evacuated 250-L tank, what is the partial pressure of nitrogen gas produced and what is the total pressure in the tank assuming the reaction has \(100 \%\) yield?

Consider a sample of a hydrocarbon (a compound consisting of only carbon and hydrogen) at \(0.959\) atm and \(298 \mathrm{~K}\). Upon combusting the entire sample in oxygen, you collect a mixture of gaseous carbon dioxide and water vapor at \(1.51\) atm and \(375 \mathrm{~K}\). This mixture has a density of \(1.391 \mathrm{~g} / \mathrm{L}\) and occupies a volume four times as large as that of the pure hydrocarbon. Determine the molecular formula of the hydrocarbon.

An ideal gas is contained in a cylinder with a volume of \(5.0 \times\) \(10^{2} \mathrm{~mL}\) at a temperature of \(30 .{ }^{\circ} \mathrm{C}\) and a pressure of 710 . torr. The gas is then compressed to a volume of \(25 \mathrm{~mL}\), and the temperature is raised to \(820 .{ }^{\circ} \mathrm{C}\). What is the new pressure of the gas?

In the "Méthode Champenoise," grape juice is fermented in a wine bottle to produce sparkling wine. The reaction is $$ \mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}(a q) \longrightarrow 2 \mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}(a q)+2 \mathrm{CO}_{2}(g) $$ Fermentation of \(750 . \mathrm{mL}\) grape juice (density \(=1.0 \mathrm{~g} / \mathrm{cm}^{3}\) ) is allowed to take place in a bottle with a total volume of \(825 \mathrm{~mL}\) until \(12 \%\) by volume is ethanol \(\left(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}\right)\). Assuming that the \(\mathrm{CO}_{2}\) is insoluble in \(\mathrm{H}_{2} \mathrm{O}\) (actually, a wrong assumption), what would be the pressure of \(\mathrm{CO}_{2}\) inside the wine bottle at \(25^{\circ} \mathrm{C}\) ? (The density of ethanol is \(0.79 \mathrm{~g} / \mathrm{cm}^{3} .\).)

In the presence of nitric acid, \(\mathrm{UO}^{2+}\) undergoes a redox process. It is converted to \(\mathrm{UO}_{2}{ }^{2+}\) and nitric oxide (NO) gas is produced according to the following unbalanced equation: \(\mathrm{H}^{+}(a q)+\mathrm{NO}_{3}^{-}(a q)+\mathrm{UO}^{2+}(a q) \longrightarrow\) \(\mathrm{NO}(g)+\mathrm{UO}_{2}^{2+}(a q)+\mathrm{H}_{2} \mathrm{O}(l)\) If \(2.55 \times 10^{2} \mathrm{~mL} \mathrm{NO}(g)\) is isolated at \(29^{\circ} \mathrm{C}\) and \(1.5 \mathrm{~atm}\), what amount (moles) of \(\mathrm{UO}^{2+}\) was used in the reaction? (Hint: Balance the reaction by the oxidation states method.)
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