Problem 38 - Chemistry: The Central Science in SI Units, Global Edition

Question

(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?

Step-by-Step Solution

Answer

(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 1: 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.

Step 2: 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\(^{-1}\): \(n = \frac{(0.003 \text{ atm})(1 \text{ L})}{(0.08206\ \text{atm L}\,\text{mol}^{-1}\,\text{K}^{-1})(250\text{ K})}\)

Step 3: 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:

Step 1: 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: \(0.040532\ \text{kPa} \times \frac{1\ \text{atm}}{101325\ \text{Pa}} = 3.998\times 10^{-4} \text{atm}\)

Step 2: Convert temperature to Kelvin

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

Step 3: 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\(^{-1}\): \(n = \frac{(3.998\times10^{-4} \text{ atm})(2 \text{ L})}{(0.08206\ \text{atm L}\,\text{mol}^{-1}\,\text{K}^{-1})(300.15\ \text{K})}\)

Step 4: Compute the number of molecules

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

Key Concepts

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.