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Chapter 20: Problem 103

The atmosphere contains \(9.0 \times 10^{-6 \%} \mathrm{Xe}\) by volume at 1.0 atm and \(25^{\circ} \mathrm{C} .\) a. Calculate the mass of \(\mathrm{Xe}\) in a room 7.26 \(\mathrm{m}\) by 8.80 \(\mathrm{m}\) by 5.67 \(\mathrm{m} .\) b. A typical person takes in about 2 \(\mathrm{L}\) of air during a breath. How many Xe atoms are inhaled in each breath?

Short Answer

Expert verified
a. The mass of Xe in the room is 0.161 g. b. The number of Xe atoms inhaled in each breath is approximately \(2.3 \times 10^{16}\) atoms.

Step by step solution

01

Calculate the volume of the room

First, find the volume of the room by multiplying its dimensions: Volume = Length × Width × Height
02

Convert the room's volume to liters

To work with the given conditions, we need to convert the room's volume from cubic meters to liters. Use the conversion factor 1 m³ = 1000 L.
03

Calculate the number of moles of air in the room

Now we'll use the ideal gas law (PV = nRT) to find the number of moles of air in the room. Rearrange the equation to solve for n (number of moles): n = PV/RT Here, we have P = 1.0 atm, V is the room's volume converted to liters, R = 0.0821 L⋅atm/mol⋅K (the ideal gas constant), and T is the temperature in Kelvin (25°C + 273.15 = 298.15 K).
04

Calculate the partial pressure of Xe

The atmosphere contains \(9.0 \times 10^{-6} \%\) of Xe by volume. To find the partial pressure of Xe, multiply the total pressure by the given percentage. Remember to convert percentage to decimal.
05

Calculate the number of moles of Xe in the room

Now, use the ideal gas law again to find the number of moles of Xe in the room. This time, use the partial pressure of Xe calculated in step 4.
06

Calculate the mass of Xe

Now, multiply the number of moles of Xe by its molar mass (131.29 g/mol) to find the mass of Xe in the room.
07

Calculate the volume of air inhaled in each breath

A typical person takes in 2 L of air during a breath. Convert the volume to liters if it's not given in liters.
08

Calculate the moles of Xe taken in one breath

Use the volume inhaled and partial pressure of Xe in the ideal gas law (PV = nRT) to find the number of moles of Xe taken in one breath.
09

Find the number of Xe atoms inhaled in each breath

To find the number of Xe atoms, multiply the number of moles of Xe taken in one breath by Avogadro's number (\(6.022 \times 10^{23} atoms/mol\)). Now you have the mass of Xe in the room and the number of Xe atoms inhaled in each breath.

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