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Chapter 8: Problem 102

The air in a flask is evacuated by a high-quality vacuum system. The vacuum created corresponds to \(1.0 \times 10^{-8}\) Torr at \(25^{\circ} \mathrm{C}\). Calculate the number of molecules of air per \(\mathrm{cm}^{3}\) remaining in the apparatus at this temperature and pressure.

Short Answer

Expert verified
Approximately \(3.24 \times 10^7\) molecules/cm\(^3\).

Step by step solution

01

Convert the Pressure to Pascals

First, we must convert the pressure from Torr to Pascals (Pa) since we typically use SI units. We know that 1 Torr is equivalent to approximately 133.322 Pa. Thus, the conversion is as follows:\[ 1.0 \times 10^{-8} \text{ Torr} \times 133.322 \frac{\text{Pa}}{\text{Torr}} = 1.33322 \times 10^{-6} \text{ Pa} \]
02

Use the Ideal Gas Law

The ideal gas law is given by: \[ PV = nRT \]where \(P\) is the pressure in Pascals, \(V\) is the volume in cubic meters, \(n\) is the number of moles of gas, \(R\) is the universal gas constant \(8.314 \text{ J/mol路K}\), and \(T\) is the temperature in Kelvin. At \(25^\circ \text{C}\), the temperature in Kelvin is \(25 + 273.15 = 298.15 \text{ K}\).
03

Calculate Number of Moles per Unit Volume

Rearrange the ideal gas law to find \(n/V\), the number of moles per cubic meter:\[ \frac{n}{V} = \frac{P}{RT} \]Substitute the known values:\[ \frac{n}{V} = \frac{1.33322 \times 10^{-6} \text{ Pa}}{8.314 \text{ J/mol路K} \times 298.15 \text{ K}} \approx 5.38 \times 10^{-11} \text{ mol/m}^3 \]
04

Convert Moles to Molecules

Use Avogadro鈥檚 number \(6.022 \times 10^{23} \text{ molecules/mol}\) to convert moles per cubic meter to molecules per cubic meter:\[ \left(5.38 \times 10^{-11} \text{ mol/m}^3\right) \times 6.022 \times 10^{23} \text{ molecules/mol} = 3.24 \times 10^{13} \text{ molecules/m}^3 \]
05

Convert Volume from Cubic Meters to Cubic Centimeters

Since we need the number of molecules per \(\text{cm}^3\), note that \(1 \text{ m}^3\) is equivalent to \(10^6 \text{ cm}^3\):\[ \frac{3.24 \times 10^{13} \text{ molecules/m}^3}{10^6 \text{ cm}^3/\text{m}^3} = 3.24 \times 10^7 \text{ molecules/cm}^3 \]

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Key Concepts

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

Pressure Conversion
When dealing with gas laws, especially in academic exercises, it鈥檚 essential to convert pressure measurements to the International System of Units (SI) for consistency. This often means converting units such as Torr into Paschals (Pa).
Torr is a unit of pressure named after the Italian scientist Evangelista Torricelli. It's commonly used in vacuum measurements. On the other hand, the Pascal is the SI unit and is used worldwide in scientific calculations.
To convert from Torr to Pascal, we use the equivalent:
- 1 Torr = 133.322 Pa.
This conversion factor allows you to seamlessly switch units and ensure calculations align properly with the requirements of the Ideal Gas Equation.
By converting 1.0 x 10鈦烩伕 Torr to Pascals, we performed the following calculation:
  • Multiply by the conversion factor:
    1.0 x 10鈦烩伕 Torr x 133.322 Pa/Torr = 1.33322 x 10鈦烩伓 Pa.
Converting pressure measurements properly is crucial for accurately using the Ideal Gas Law to solve for quantities such as the number of molecules or moles of gas.
Moles to Molecules Conversion
Once you have found the number of moles using the Ideal Gas Law, the next step is often to convert this value to the number of individual molecules. This is where Avogadro鈥檚 number comes into play.
Avogadro's number (\(6.022 imes 10^{23} ext{ molecules/mol}\)) is a constant that tells you how many molecules are present in exactly one mole of any substance.
To convert from moles to molecules, simply multiply the number of moles by Avogadro's number.
  • In our example, an initial calculation revealed \(5.38 imes 10^{-11} ext{ mol/m}^3\).
    To convert to molecules: \[(5.38 imes 10^{-11} ext{ mol/m}^3) imes 6.022 imes 10^{23} ext{ molecules/mol} = 3.24 imes 10^{13} ext{ molecules/m}^3\]
Utilizing Avogadro's number is key when you need to transition from bulk measurement (moles) to discrete counts (molecules), aiding in understanding the actual quantity of particles in a given volume.
Volume Conversion
In problems involving gases, you may need to convert the volume to a different unit to make your answers more intuitive or to match the requirements of the problem. This kind of conversion ensures consistency across calculations.
For our problem, we needed to express the number of molecules in cubic centimeters (\( ext{cm}^3\)) rather than cubic meters (\( ext{m}^3\)), which is the SI unit for volume. Since 1 \( ext{m}^3 = 10^6 ext{ cm}^3\), you can convert by dividing by that factor:
  • The calculation here takes the number of molecules per cubic meter and adjusts it to per cubic centimeter: \[\frac{3.24 imes 10^{13} ext{ molecules/m}^3}{10^6 ext{ cm}^3/ ext{m}^3} = 3.24 imes 10^7 ext{ molecules/cm}^3\]
Volume conversion ensures that the scale of the numbers you鈥檙e working with remains practical and within the context of the problem, helping to maintain accuracy and understandability.

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Most popular questions from this chapter

The density of air at \(20.0 \mathrm{~km}\) above Earth's surface is \(92 \mathrm{~g} / \mathrm{m}^{3}\). The pressure is \(42 \mathrm{mmHg}\) and the temperature is \(-63^{\circ}\) C. Assuming the atmosphere contains only \(\mathrm{O}_{2}\) and \(\mathrm{N}_{2},\) calculate (a) the average molar mass of the air at \(20.0 \mathrm{~km}\). (b) the mole fraction of each gas.

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