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Chapter 5: Problem 60

An ideal gas at \(7^{\circ} \mathrm{C}\) is in a spherical flexible container having a radius of \(1.00 \mathrm{~cm}\). The gas is heated at constant pressure to \(88^{\circ} \mathrm{C}\). Determine the radius of the spherical container after the gas is heated. [Volume of a sphere \(\left.=(4 / 3) \pi r^{3} .\right]\)

Short Answer

Expert verified
The radius of the spherical container after the gas is heated is approximately \(1.093 \, \mathrm{cm}\).

Step by step solution

01

Convert temperatures to Kelvin

In order to use the ideal gas law, we need to convert the given temperatures from Celsius to Kelvin. We can do this by adding 273.15 to the Celsius temperature. Initial temperature: \(T_1 = 7 + 273.15 = 280.15\) K Final temperature: \(T_2 = 88 + 273.15 = 361.15\) K
02

Calculate initial volume

Given the initial radius, we can compute the initial volume of the gas using the spherical volume formula provided in the exercise. Initial volume: \(V_1 = (4/3) \pi (1.00)^3 = \dfrac{4}{3} \pi\) cm³
03

Use the ideal gas law to determine the volume ratio

As the pressure and number of moles remain constant during the process, we can say: \(\dfrac{V_1}{T_1} = \dfrac{V_2}{T_2}\) We can write the equation in terms of V_2, \( V_2 = V_1 \frac{T_2}{T_1}\)
04

Calculate final volume

Now that we have the volume ratio, we can calculate the final volume after the heating process. \(V_2 = \dfrac{4}{3} \pi \dfrac{361.15}{280.15} \approx 1.289 \cdot \dfrac{4}{3}\pi\) cm³
05

Calculate the final radius

Now we can use the volume formula for a sphere to find the final radius of the container. \(V_2 = \dfrac{4}{3} \pi r_2^3\) Solve it for r_2: \(r_2^3 = \dfrac{3 V_2}{4 \pi} \approx 1.289\) \(r_2 = \sqrt[3]{1.289} \approx 1.093\) cm The radius of the spherical container after the gas is heated is approximately 1.093 cm.

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Most popular questions from this chapter

Calculate the pressure exerted by \(0.5000\) mole of \(\mathrm{N}_{2}\) in a 10.000-L container at \(25.0^{\circ} \mathrm{C}\) a. using the ideal gas law. b. using the van der Waals equation. c. Compare the results. d. Compare the results with those in Exercise 115 .

A \(15.0\) -L rigid container was charged with \(0.500\) atm of krypton gas and \(1.50\) atm of chlorine gas at \(350 .{ }^{\circ} \mathrm{C}\). The krypton and chlorine react to form krypton tetrachloride. What mass of krypton tetrachloride can be produced assuming \(100 \%\) yield?

Consider two gases, A and B, each in a 1.0-L container with both gases at the same temperature and pressure. The mass of gas \(\mathrm{A}\) in the container is \(0.34 \mathrm{~g}\) and the mass of gas \(\mathrm{B}\) in the container is \(0.48 \mathrm{~g}\). a. Which gas sample has the most molecules present? Explain. b. Which gas sample has the largest average kinetic energy? Explain. c. Which gas sample has the fastest average velocity? Explain. d. How can the pressure in the two containers be equal to each other since the larger gas B molecules collide with the container walls more forcefully?

Consider separate \(1.0\) -L gaseous samples of \(\mathrm{He}, \mathrm{N}_{2}\), and \(\mathrm{O}_{2}\), all at STP and all acting ideally. Rank the gases in order of increasing average kinetic energy and in order of increasing average velocity.

A 20.0-L stainless steel container at \(25^{\circ} \mathrm{C}\) was charged with \(2.00\) atm of hydrogen gas and \(3.00\) atm of oxygen gas. A spark ignited the mixture, producing water. What is the pressure in the tank at \(25^{\circ} \mathrm{C} ?\) If the exact same experiment were performed, but the temperature was \(125^{\circ} \mathrm{C}\) instead of \(25^{\circ} \mathrm{C}\), what would be the pressure in the tank?
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