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Chapter 6: Problem 15

A weather balloon filled with He gas has a volume of \(2.00 \times 10^{3} \mathrm{m}^{3}\) at ground level, where the atmospheric pressure is 1.000 atm and the temperature \(27^{\circ}\) C. After the balloon rises high above Earth to a point where the atmospheric pressure is 0.340 atm, its volume increases to \(5.00 \times 10^{3} \mathrm{m}^{3} .\) What is the temperature of the atmosphere at this altitude?

Short Answer

Expert verified
The temperature of the atmosphere at the altitude where the atmospheric pressure is 0.34 atm is approximately \( -18.04^{\circ}C\)

Step by step solution

01

Initial Temperature Conversion

Convert the initial temperature from Celsius to Kelvin by using the conversion equation \(K = C + 273.15\). Thus, the initial temperature in Kelvin is \(27^{\circ}C + 273.15K = 300.15K\).
02

Apply the Ideal Gas Law

Since the number of moles and the gas constant don't change, and we can equate the initial and final states of the balloon using the equation \(P_1V_1/T_1 = P_2V_2/T_2\), where 1 refers to the initial state and 2 refers to the final.
03

Solve for Final Temperature

We can rearrange the equation from step 2 to solve for \(T_2\) giving us the equation \(T_2 = P_2V_2T_1/(P_1V_1)\). Substituting the given values into this equation gives us the final temperature in Kelvin as \(T_2 = (0.340\, atm \times 5000\,m^3 \times 300.15\,K) / (1.000\, atm \times 2000\,m^3) = 255.1125\,K\).
04

Final Temperature Conversion

Convert the final temperature from Kelvin back to Celsius using the conversion equation \(C = K - 273.15\), giving a final temperature of \(255.1125\,K - 273.15 = -18.04^{\circ}C.\)

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Most popular questions from this chapter

What is the pressure (in \(\mathrm{mmHg}\) ) of the gas inside the apparatus below if \(P_{\text {bar. }}=740 \mathrm{mm} \mathrm{Hg}, h_{1}=30 \mathrm{mm}\) and \(h_{2}=50 \mathrm{mm} ?\)

A 3.05 g sample of \(\mathrm{NH}_{4} \mathrm{NO}_{3}(\mathrm{s})\) is introduced into an evacuated 2.18 L flask and then heated to \(250^{\circ} \mathrm{C}\).What is the total gas pressure, in atmospheres, in the flask at \(250^{\circ} \mathrm{C}\) when the \(\mathrm{NH}_{4} \mathrm{NO}_{3}\) has completely decomposed? $$\mathrm{NH}_{4} \mathrm{NO}_{3}(\mathrm{s}) \longrightarrow \mathrm{N}_{2} \mathrm{O}(\mathrm{g})+2 \mathrm{H}_{2} \mathrm{O}(\mathrm{g})$$

A sounding balloon is a rubber bag filled with \(\mathrm{H}_{2}(\mathrm{g})\) and carrying a set of instruments (the payload). Because this combination of bag, gas, and payload has a smaller mass than a corresponding volume of air, the balloon rises. As the balloon rises, it expands. From the table below, estimate the maximum height to which a spherical balloon can rise given the mass of balloon, \(1200 \mathrm{g} ;\) payload, \(1700 \mathrm{g}\) : quantity of \(\mathrm{H}_{2}(\mathrm{g})\) in balloon, \(120 \mathrm{ft}^{3}\) at \(0.00^{\circ} \mathrm{C}\) and \(1.00 \mathrm{atm}\); diameter of balloon at maximum height, 25 ft. Air pressure and temperature as functions of altitude are: $$\begin{array}{ccl} \hline \text { Altitude, km } & \text { Pressure, mb } & \text { Temperature, } \mathrm{K} \\ \hline 0 & 1.0 \times 10^{3} & 288 \\ 5 & 5.4 \times 10^{2} & 256 \\ 10 & 2.7 \times 10^{2} & 223 \\ 20 & 5.5 \times 10^{1} & 217 \\ 30 & 1.2 \times 10^{1} & 230 \\ 40 & 2.9 \times 10^{0} & 250 \\ 50 & 8.1 \times 10^{-1} & 250 \\ 60 & 2.3 \times 10^{-1} & 256 \\ \hline \end{array}$$

A 2.35 L container of \(\mathrm{H}_{2}(\mathrm{g})\) at \(762 \mathrm{mmHg}\) and \(24^{\circ} \mathrm{C}\) is connected to a 3.17 L container of \(\mathrm{He}(\mathrm{g})\) at \(728 \mathrm{mmHg}\) and \(24^{\circ} \mathrm{C}\). After mixing, what is the total gas pressure, in millimeters of mercury, with the temperature remaining at \(24^{\circ} \mathrm{C} ?\)

A 2.00 L container is filled with \(\operatorname{Ar}(g)\) at 752 mm Hg and \(35^{\circ} \mathrm{C} .\) A \(0.728 \mathrm{g}\) sample of \(\mathrm{C}_{6} \mathrm{H}_{6}\) vapor is then added. (a) What is the total pressure in the container? (b) What is the partial pressure of \(\mathrm{Ar}\) and of \(\mathrm{C}_{6} \mathrm{H}_{6} ?\)
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