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Chapter 10: Problem 38

(a) If the pressure exerted by ozone, \(\mathrm{O}_{3}\), in the stratosphere is \(304 \mathrm{~Pa}\) and the temperature is \(250 \mathrm{~K}\), how many ozone molecules are in a liter? (b) Carbon dioxide makes up approximately \(0.04 \%\) of Earth's atmosphere. If you collect a 2.0-L sample from the atmosphere at sea level \((101.33 \mathrm{kPa})\) on a warm day \(\left(27^{\circ} \mathrm{C}\right)\), how many \(\mathrm{CO}_{2}\) molecules are in your sample?

Short Answer

Expert verified
(a) The number of ozone molecules in a liter is: \(n_{molecules} = n \times (6.022\times 10^{23}\ \text{molecules/mol})\) where \(n = \frac{(0.003 \text{ atm})(1 \text{ L})}{(0.08206\ \text{atm L}\,\text{mol}^{-1}\,\text{K}^{-1})(250\text{ K})}\) (b) The number of CO2 molecules in a 2.0-L sample is: \(n_{molecules} = n \times (6.022\times 10^{23}\ \text{molecules/mol})\) where \(n = \frac{(3.998\times10^{-4} \text{ atm})(2 \text{ L})}{(0.08206\ \text{atm L}\,\text{mol}^{-1}\,\text{K}^{-1})(300.15\ \text{K})}\)

Step by step solution

01

Convert pressure to atm and volume to liters

: We are given the pressure in Pascals and the temperature in Kelvin. Convert the pressure to atm: \(\frac{1\ \text{atm}}{101325\ \text{Pa}} \times 304\ \text{Pa} = 0.003 \text{atm}\) The volume is given in liters, which is 1 L.
02

Calculate number of moles (n)

: Rearrange the ideal gas law equation for n: \(n = \frac{PV}{RT}\) Plugging in the given values and the gas constant (R) = 0.08206 atm L mol\(^{-1}\) K\(^{-1}\): \(n = \frac{(0.003 \text{ atm})(1 \text{ L})}{(0.08206\ \text{atm L}\,\text{mol}^{-1}\,\text{K}^{-1})(250\text{ K})}\)
03

Compute the number of molecules

: Calculate the number of ozone molecules by multiplying the number of moles by Avogadro's number: \(n_{molecules} = n \times (6.022\times 10^{23}\ \text{molecules/mol})\) (b) Number of CO2 molecules in a 2.0-L sample:
04

Calculate the partial pressure of CO2

: Given that carbon dioxide makes up approximately 0.04% of Earth's atmosphere, calculate the partial pressure of CO2 by multiplying the percentage by the total atmospheric pressure: \((0.0004) \times 101.33\ \text{kPa} = 0.040532\ \text{kPa} \) Convert this partial pressure from kPa to atm: \(0.040532\ \text{kPa} \times \frac{1\ \text{atm}}{101325\ \text{Pa}} = 3.998\times 10^{-4} \text{atm}\)
05

Convert temperature to Kelvin

: Given the temperature in Celsius, convert it to Kelvin: \(27 ^{\circ}\ \text{C} + 273.15\ = 300.15\ \text{K}\)
06

Calculate number of moles (n)

: Rearrange the ideal gas law equation for n: \(n = \frac{PV}{RT}\) Plugging in the given values and R = 0.08206 atm L mol\(^{-1}\) K\(^{-1}\): \(n = \frac{(3.998\times10^{-4} \text{ atm})(2 \text{ L})}{(0.08206\ \text{atm L}\,\text{mol}^{-1}\,\text{K}^{-1})(300.15\ \text{K})}\)
07

Compute the number of molecules

: Calculate the number of CO2 molecules by multiplying the number of moles by Avogadro's number: \(n_{molecules} = n \times (6.022\times 10^{23}\ \text{molecules/mol})\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Avogadro's Number
Avogadro's Number is crucial in understanding chemical reactions and compositions. It represents the number of constituent particles, usually atoms or molecules, contained in one mole of a substance. This scientific constant was named after Amedeo Avogadro, an Italian scientist. The value of Avogadro's Number is
  • 6.022 x 1023 molecules/mol
This immense number helps chemists convert between microscopic models to macroscopic scales, understanding how many molecules are present in a given sample.
For instance, in the exercise, when we calculate the number of ozone or carbon dioxide molecules, we multiply the number of moles by Avogadro's Number to determine the absolute number of molecules present. This conversion is essential because it translates moles, which are abstract in nature, into a concrete quantity of atoms or molecules.
Partial Pressure
Partial pressure is a critical concept in gas laws, especially when dealing with mixtures of gases. It refers to the pressure that a single gas in a mixture would exert if it occupied the entire volume alone. This idea allows us to understand the behavior of individual gas components within a mixture.
In the step-by-step solution, partial pressure is calculated for carbon dioxide (CO2) by considering its percentage of the atmosphere and the total atmospheric pressure. The formula used is:
  • Partial Pressure of CO2 = Total Pressure x Mole Fraction of CO2
Calculating partial pressure is invaluable because it enables us to focus on a single gaseous component in a system without interference from other gases. This is immensely beneficial in understanding behaviors and interactions in chemical systems.
Moles Calculation
The concept of moles calculation is integral to chemical analysis, allowing chemists to quantify substances accurately. A mole is defined as a chemical mass unit, equal to 6.022 x 1023 molecules, atoms, or other elementary units, known as Avogadro's Number.

Utilizing the Ideal Gas Law

Moles can be determined using the Ideal Gas Law, simplified to the equation \( n = \frac{PV}{RT} \). Here, \( n \) is the number of moles, \( P \) is pressure, \( V \) is volume, \( R \) is the ideal gas constant, and \( T \) is temperature in Kelvin.
In the exercises given, the ideal gas law is rearranged to solve for \( n \), shedding light on the number of moles contained within a specified volume of gas at varying conditions of pressure and temperature.
  • Step 1: Ensure all units are in the correct SI format - pressure in atm, volume in liters, and temperature in Kelvin.
  • Step 2: Substitute the known values into the rearranged equation to find the moles.
  • Step 3: Use Avogadro's Number to convert moles into molecules.
Understanding this process is fundamental in tracking the quantity of any gaseous substance and predicting how it will interact or transform in reactions.

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Most popular questions from this chapter

A sample of \(3.00 \mathrm{~g}\) of \(\mathrm{SO}_{2}(g)\) originally in a \(5.00-\mathrm{L}\) vessel at \(21{ }^{\circ} \mathrm{C}\) is transferred to a \(10.0-\mathrm{L}\) vessel at \(26^{\circ} \mathrm{C}\). A sample of \(2.35 \mathrm{~g}\) of \(\mathrm{N}_{2}(g)\) originally in a \(2.50-\mathrm{L}\) vessel at \(20^{\circ} \mathrm{C}\) is transferred to this same \(10.0-\mathrm{L}\) vessel. \((\mathbf{a})\) What is the partial pressure of \(\mathrm{SO}_{2}(g)\) in the larger container? (b) What is the partial pressure of \(\mathrm{N}_{2}(g)\) in this vessel? (c) What is the total pressure in the vessel?

Which of the following statements best explains why a closed balloon filled with helium gas rises in air? (a) Helium is a monatomic gas, whereas nearly all the molecules that make up air, such as nitrogen and oxygen, are diatomic. (b) The average speed of helium atoms is greater than the average speed of air molecules, and the greater speed of collisions with the balloon walls propels the balloon upward. (c) Because the helium atoms are of lower mass than the average air molecule, the helium gas is less dense than air. The mass of the balloon is thus less than the mass of the air displaced by its volume. (d) Because helium has a lower molar mass than the average air molecule, the helium atoms are in faster motion. This means that the temperature of the helium is greater than the air temperature. Hot gases tend to rise.

The highest barometric pressure ever recorded was 823.7 torr at Agata in Siberia, Russia on December 31,1968 . Convert this pressure to (a) atm, (b) \(\mathrm{mm} \mathrm{Hg}\), (c) pascals, (d) bars, (e) psi.

Rank the following gases from least dense to most dense at \(101.33 \mathrm{kPa}\) and \(298 \mathrm{~K}: \mathrm{O}_{2}, \mathrm{Ar}, \mathrm{NH}_{3}, \mathrm{HCl}\).

Assume that a single cylinder of an automobile engine has a volume of \(524 \mathrm{~cm}^{3}\). (a) If the cylinder is full of air at \(74^{\circ} \mathrm{C}\) and \(99.3 \mathrm{kPa}\), how many moles of \(\mathrm{O}_{2}\) are present? (The mole fraction of \(\mathrm{O}_{2}\) in dry air is \(0.2095 .\) ) (b) How many grams of \(\mathrm{C}_{8} \mathrm{H}_{18}\) could be combusted by this quantity of \(\mathrm{O}_{2}\), assuming complete combustion with formation of \(\mathrm{CO}_{2}\) and \(\mathrm{H}_{2} \mathrm{O} ?\)
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