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Chapter 18: Problem 80

The estimated average concentration of \(\mathrm{NO}_{2}\) in air in the United States in 2006 was 0.016 ppm. (a) Calculate the partial pressure of the \(\mathrm{NO}_{2}\) in a sample of this air when the atmospheric pressure is 755 torr \((99.1 \mathrm{kPa})\). (b) How many molecules of \(\mathrm{NO}_{2}\) are present under these conditions at \(20{ }^{\circ} \mathrm{C}\) in a room that measures \(15 \times 14 \times 8 \mathrm{ft}\) ?

Short Answer

Expert verified
The number of molecules of NO2 in the room under the given conditions is approximately \( 1.91 \times 10^{19} \) molecules.

Step by step solution

01

Convert the concentration of NO2 from ppm to a fraction

To convert the concentration of NO2 from ppm to a fraction, divide the concentration (0.016 ppm) by 1,000,000 (since ppm stands for parts per million): \[ \frac{0.016}{1,000,000} = 1.6 \times 10^{-8} \]
02

Calculate the partial pressure of NO2

To calculate the partial pressure of NO2, multiply the atmospheric pressure (755 torr) by the fraction obtained in Step 1: \[ 755 \text{ torr} \times 1.6 \times 10^{-8} = 1.21 \times 10^{-5}\text{ torr} \]
03

Find the number of moles of air in the room

We can use the ideal gas law (PV = nRT) to find the number of moles of air in the room. First, we need to convert the room dimensions from feet to meters: \[ \frac{15 \text{ ft} \times 14 \text{ ft} \times 8 \text{ ft}}{3.2808^3\frac{\text{m}^3}{\text{ft}^3}} = 47.89 \text{ m}^3 \] Next, we'll convert the pressure from torr to Pa: \[ \frac{755 \text{ torr}}{0.007501 \frac{\text{torr}}{\text{Pa}}} = 100662.86 \text{ Pa} \] Now, we'll plug our values into the ideal gas law, rearranged for the number of moles: \[ n = \frac{PV}{RT} = \frac{ (100662.86 \text{ Pa})(47.89 \text{ m}^3)}{(8.314 \text{ J/(mol·K)})(20 + 273.15 \text{ K})} = 1984.99 \text{ moles} \]
04

Calculate the number of moles of NO2 in the room

To find the number of moles of NO2, multiply the fraction obtained in Step 1 by the total number of moles of air in the room: \[ 1.6 \times 10^{-8} \times 1984.99 \text{ moles} = 3.18 \times 10^{-5} \text{ moles of } NO_2 \]
05

Convert the moles of NO2 to molecules of NO2

To convert the moles of NO2 to molecules, multiply the number of moles by Avogadro's number (6.022 x 10^23 molecules/mol): \[ 3.18 \times 10^{-5} \text{ moles of } NO_2 \times 6.022 \times 10^{23} \frac{\text{molecules}}{\text{mol}} = 1.91 \times 10^{19} \text{ molecules of } NO_2 \] The number of molecules of NO2 in the room under the given conditions is approximately \( 1.91 \times 10^{19} \) molecules.

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