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Chapter 5: Problem 45

Ozone molecules in the stratosphere absorb much of the harmful radiation from the sun. Typically, the temperature and pressure of ozone in the stratosphere are \(250 \mathrm{~K}\) and \(1.0 \times 10^{-3}\) atm, respectively. How many ozone molecules are present in \(1.0 \mathrm{~L}\) of air under these conditions?

Short Answer

Expert verified
There are \(2.93 \times 10^{21}\) ozone molecules in \(1.0 \mathrm{~L}\) of air under these conditions.

Step by step solution

01

Convert the Pressure into SI units

The pressure is given in atm, but in the Ideal Gas Law equation, the pressure should be in Pascals (Pa). The pressure is \(1.0 \times 10^{-3}\) atm, which can be converted to Pa using the conversion factor \(1 \mathrm{~atm} = 1.01325 \times 10^{5} \mathrm{~Pa}\). So the pressure in Pa is \(1.0 \times 10^{-3} \times 1.01325 \times 10^{5} = 101.325 \mathrm{~Pa}\).
02

Use the Ideal Gas Law to calculate the number of mol

Rearrange the Ideal Gas Law, so that \(n = \frac{PV}{RT}\). Then plug in the given values. The volume \(V = 1.0 \mathrm{~L} = 1.0 \times 10^{-3} \mathrm{~m^3}\) (remember to convert the volume from L to m^3, 1 L = 0.001 m^3), the gas constant \(R = 8.314 \mathrm{~J/K mol}\), and the temperature \(T = 250\mathrm{~K}\). Solving this equation gives \(n = \frac{(101.325)(1.0 \times 10^{-3})}{(8.314)(250)} = 4.87 \times 10^{-3} \mathrm{~mol}\).
03

Convert the number of mol to the number of molecules

Use Avogadro's number to convert mol to molecules. \(1 \mathrm{~mol}\) contains \(6.022 \times 10^{23}\) molecules. Therefore, the number of ozone molecules is \(4.87 \times 10^{-3} \times 6.022 \times 10^{23} = 2.93 \times 10^{21} \mathrm{~molecules}\).

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