Question

Determine the excluded volume per mole and the total volume of the molecules in a mole for a gas consisting of molecules with radius 165 picometers (pm). [Note: To obtain the volume in liters, we must express the radius in decimeters (dm).]

Short answer

Convert, Calculate, Multiply, and Adjust for 4 times to find excluded volume.

Step-by-step solution

Convert Radius to Decimeters

First, convert the given radius from picometers to decimeters. Since 1 picometer is 1x10^{-12} meters and 1 meter is 10 decimeters, we have: \[165 ext{ pm} = 165 imes 10^{-12} ext{ m} = 165 imes 10^{-13} ext{ dm}\]

Calculate the Volume of a Single Molecule

The volume of a sphere is given by the formula \( V = \frac{4}{3} \pi r^3 \). Using the converted radius in decimeters: \[ V = \frac{4}{3} \pi (165 imes 10^{-13})^3 \text{ dm}^3 \]

Calculate Volume per Mole for the Molecules

To find the total volume for one mole, multiply the volume of one molecule by Avogadro's number \(6.022 \times 10^{23}\) molecules per mole: \[V_{total} = V \times 6.022 \times 10^{23}\]

Determine Excluded Volume per Mole

The excluded volume per molecule is a measure of the space taken up by a single molecule. For gases, the excluded volume is typically 4 times the actual volume calculated earlier. Thus, for a mole of molecules:\[ V_{excluded} = 4 \times V_{total}\]

Answer Calculation

Now, compute using the numbers: first calculate the volume of a single molecule from step 2, then use it in step 3, and finally step 4. Substitute numerical values to get the excluded volume per mole and the total volume per mole.

Key concepts

Volume of Sphere

The volume of a sphere is essential in understanding how much space an object takes up in three dimensions. The formula to calculate the volume of a sphere is given by:\[ V = \frac{4}{3} \pi r^3 \]In this formula, \( V \) represents the volume, \( r \) is the radius, and \( \pi \) (approximately 3.14159) is a constant.To accurately use this formula, ensure your radius is in the correct units, such as meters or decimeters. In our exercise, the radius of the molecule given was in picometers and needed conversion to decimeters before applying the volume formula.The sphere volume formula is crucial because it gives insight into the size of molecules, which is foundational for calculating molecular properties like the excluded volume.

Avogadro's Number

Avogadro's number is a key concept when discussing quantities of particles at the atomic and molecular scale. It is approximately \( 6.022 \times 10^{23} \) and represents the number of atoms, ions, or molecules in one mole of a substance.This number allows chemists to bridge the gap between the macroscopic and microscopic worlds. By using Avogadro's number, one can calculate the total volume for a large number of molecules, like a mole.

• This makes it possible to convert individual molecular volumes into practical quantities.
• In our problem, applying Avogadro's number helped find the total volume a mole of molecules occupies.

It acts as a scaling factor from an individual molecule's volume to a mole's collective volume.

Molecular Volume

Molecular volume refers to the space a molecule occupies. Essentially, it's the volume calculated for a single molecule—the starting point is the volume of a sphere, as many molecules can be approximated as spheres.Once you've computed the single molecule's volume using the sphere formula, you can find the molecular volume of one mole of molecules by multiplying by Avogadro's number.This concept helps us appreciate just how small molecules are since even a large number, like a mole (\( 6.022 \times 10^{23} \)), results in a total volume that is manageable in scientific experiments and calculations.

Radius Conversion

Radius conversion is a critical step in scientific calculations, especially when working with units at the molecular scale. In the original exercise, we converted the radius from picometers to decimeters.

• Conversions are vital to ensure the consistency of units, especially when using formulas that require specific units.
• The conversion factor in this exercise involved multiple steps, first changing picometers to meters, then to decimeters: \(1 \text{ pm} = 1 \times 10^{-12} \text{ meters} = 1 \times 10^{-13} \text{ dm} \).

This correct unit conversion ensures that the volume calculation is accurate, thereby leading to reliable results for subsequent steps like determining excluded and molecular volumes.