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Chapter 10: Problem 77

Radon (Rn) is the heaviest (and only radioactive) member of the noble gases. How much slower is the root-mean-square speed of Rn than He at $300 \mathrm{K?}$

Short Answer

Expert verified
The root-mean-square speed of Radon (Rn) is approximately 1.17 x 10^3 m/s slower than the root-mean-square speed of Helium (He) at 300 K.

Step by step solution

01

Identify the given values and constants

Given values: Temperature (T) = 300 K Constants: Boltzmann constant (k_B) = 1.38 × 10^-23 J/K
02

Find the molar mass of both Helium and Radon

Molar mass of He = 4 g/mol Molar mass of Rn = 222 g/mol
03

Convert molar mass to molecular mass

To convert molar mass to molecular mass, we need to multiply molar mass by the unified atomic mass unit (u). 1 u = 1.66 × 10^-27 kg Molecular mass of He = (4 g/mol) * (1.66 × 10^-27 kg/u) = 6.64 × 10^-27 kg Molecular mass of Rn = (222 g/mol) * (1.66 × 10^-27 kg/u) = 3.68 × 10^-25 kg
04

Calculate the root-mean-square speeds for both Helium and Radon

Now we can plug these values into the root-mean-square formula: \(v_{rms, He} = \sqrt{\frac{3 * 1.38 * 10^{-23} * 300}{6.64 * 10^{-27}}}\) \(v_{rms, He} ≈ 1.38 * 10^3 m/s\) \(v_{rms, Rn} = \sqrt{\frac{3 * 1.38 * 10^{-23} * 300}{3.68 * 10^{-25}}}\) \(v_{rms, Rn} ≈ 2.14 * 10^2 m/s\)
05

Find the difference in root mean square speeds

Now we will find the difference in root mean square speeds of He and Rn: Difference = Speed of He - Speed of Rn Difference = \(1.38 * 10^3 - 2.14 * 10^2 ≈ 1.17 * 10^3 m/s\) The root-mean-square speed of Radon (Rn) is approximately 1.17 x 10^3 m/s slower than the root-mean-square speed of Helium (He) at 300 K.

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Most popular questions from this chapter

Magnesium can be used as a "getter" in evacuated enclosures to react with the last traces of oxygen. (The magnesium is usually heated by passing an electric current through a wire or ribbon of the metal.) If an enclosure of $5.67 \mathrm{~L}\( has a partial pressure of \)\mathrm{O}_{2}\( of \)7.066 \mathrm{mPa}\( at \)30^{\circ} \mathrm{C}$, what mass of magnesium will react according to the following equation? $$2 \mathrm{Mg}(s)+\mathrm{O}_{2}(g) \longrightarrow 2 \mathrm{MgO}(s).$$

Ammonia and hydrogen chloride react to form solid ammonium chloride: $$\mathrm{NH}_{3}(g)+\mathrm{HCl}(g) \longrightarrow \mathrm{NH}_{4} \mathrm{Cl}(s)$$ Two \(2.00-\mathrm{L}\) flasks at \(25^{\circ} \mathrm{C}\) are connected by a valve, as shown in the drawing. One flask contains \(5.00 \mathrm{~g}\) of \(\mathrm{NH}_{3}(g),\) and the other contains \(5.00 \mathrm{~g}\) of \(\mathrm{HCl}(g) .\) When the valve is opened, the gases react until one is completely consumed. (a) Which gas will remain in the system after the reaction is complete? (b) What will be the final pressure of the system after the reaction is complete? (Neglect the volume of the ammonium chloride formed.) (c) What mass of ammonium chloride will be formed?

A plasma-screen TV contains thousands of tiny cells filled with a mixture of Xe, Ne, and He gases that emits light of specific wavelengths when a voltage is applied. A particular plasma cell, $0.900 \mathrm{~mm} \times 0.300 \mathrm{~mm} \times 10.0 \mathrm{~mm},\( contains \)4 \%$ Xe in a 1: 1 Ne:He mixture at a total pressure of \(66.66 \mathrm{kPa}\). Calculate the number of Xe, Ne, and He atoms in the cell and state the assumptions you need to make in your calculation.

Does the effect of intermolecular attraction on the properties of a gas become more significant or less significant if (a) the gas is compressed to a smaller volume at constant temperature; \((\mathbf{b})\) the temperature of the gas is increased at constant volume?

(a) The compound 1 -iodododecane is a nonvolatile liquid with a density of \(1.20 \mathrm{~g} / \mathrm{mL}\). The density of mercury is $13.6 \mathrm{~g} / \mathrm{mL}$. What do you predict for the height of a barometer column based on 1 -iodododecane, when the atmospheric pressure is 749 torr? \((\mathbf{b})\) What is the pressure, in atmospheres, on the body of a diver if he is $21 \mathrm{ft}$ below the surface of the water when the atmospheric pressure is 742 torr?
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