Question

What volume (in liters) is occupied by \(3.38 \times 10^{22}\) nitrogen molecules at \(100^{\circ} \mathrm{C}\) and \(255 \mathrm{~mm} \mathrm{Hg} ?\)

Short answer

The volume occupied is approximately 5.11 liters.

Step-by-step solution

Convert Pressure to Atmospheres

Pressure is given in millimeters of mercury (mm Hg). Firstly, convert this to atmospheres as it's a standard unit in gas law calculations. Use the conversion factor: \[1 ext{ atm} = 760 ext{ mm Hg}\] Thus, \[P = \frac{255 ext{ mm Hg}}{760 ext{ mm Hg/atm}} = 0.3355 ext{ atm}\]

Convert Temperature to Kelvin

Temperature is given in degrees Celsius. Convert it to Kelvin by adding 273.15 to the Celsius temperature: \[T = 100^{\circ}C + 273.15 = 373.15 ext{ K}\]

Calculate Moles of Gas

Use Avogadro's number to convert molecules to moles: Avogadro's Number = \(6.022 \times 10^{23} \text{ molecules/mol}\).Number of moles \(n\) is calculated as: \[n = \frac{3.38 \times 10^{22} \text{ molecules}}{6.022 \times 10^{23} \text{ molecules/mol}} \approx 0.0561 \text{ mol}\]

Use the Ideal Gas Law

The ideal gas law is \[PV = nRT\]where \(R\) is the ideal gas constant, \(0.0821 ext{ L atm/mol K}\). Substitute the known values into the equation:\[V = \frac{nRT}{P} = \frac{0.0561 \text{ mol} \times 0.0821 \text{ L atm/mol K} \times 373.15 \text{ K}}{0.3355 \text{ atm}}\]

Solve for Volume

Calculate the volume using the values from the previous steps:\[V \approx \frac{0.0561 \times 0.0821 \times 373.15}{0.3355} \approx 5.11 \text{ liters}\]

Key concepts

Pressure Conversion

Pressure is a critical component when utilizing the ideal gas law, and it's often necessary to convert pressure units to ensure consistency. In many exercises, like in this case, pressure is provided in millimeters of mercury (mm Hg). However, the ideal gas law formula typically requires pressure in atmospheres (atm).

To convert mm Hg to atm, use the conversion factor:

• 1 atm = 760 mm Hg

By dividing the given pressure in mm Hg by 760 mm Hg/atm, you convert the pressure to atmospheres. For instance, if you have a pressure of 255 mm Hg, the conversion would look like:

• \( P = \frac{255 \text{ mm Hg}}{760 \text{ mm Hg/atm}} \approx 0.3355 \text{ atm} \)

This conversion ensures that the pressure units are compatible with the ideal gas constant when calculating volume.

Temperature Conversion

In gas law calculations, temperature must always be expressed in Kelvin. Kelvin is the absolute temperature scale and ensures that calculations remain consistent across various applications.

To convert Celsius to Kelvin, you simply add 273.15 to the temperature in degrees Celsius. This is because the Kelvin scale starts at absolute zero, which is approximately -273.15°C. For instance:

• If the temperature is 100°C, then the conversion to Kelvin is: \( T = 100^{\circ}C + 273.15 = 373.15 \text{ K} \)

Converting to Kelvin before using the ideal gas law ensures that temperatures are in a format that works with the gas constant, making your calculations accurate.

Moles Calculation

Calculating the moles of a substance is crucial when using the ideal gas law. It allows one to connect the amount of gas to its volume, pressure, and temperature.

Avogadro's number provides the link between the number of molecules and moles. Avogadro’s number is a scientific constant with a value of:

• \( 6.022 \times 10^{23} \text{ molecules/mol} \)

By dividing the number of molecules you have by Avogadro's number, you obtain the amount in moles. For example:

• If you have \( 3.38 \times 10^{22} \text{ molecules} \), the number of moles \( n \) is: \( n = \frac{3.38 \times 10^{22}}{6.022 \times 10^{23}} \approx 0.0561 \text{ mol} \)

This conversion aids in using the ideal gas law to find other properties like volume.

Avogadro's Number

Avogadro's number is a fundamental constant in chemistry and plays a crucial role in molecular and molar conversions. It denotes the number of particles – be they atoms, molecules, or ions – in one mole of a substance.

Avogadro's number is:

• \( 6.022 \times 10^{23} \text{ particles/mol} \)

This constant helps convert a certain number of molecules into moles, providing a way to connect the microscopic world of atoms and molecules with the macroscopic quantities measured in the lab. For example, with Avogadro’s number, you can take a known amount of molecules and determine how many moles they represent.By converting to moles, we can then use equations like the ideal gas law to make practical calculations about gases under various conditions.