Question

A \(0.100 \ell\) container maintained at constant temperature contains \(5.0 \times 10^{10}\) molecules of an ideal gas. How many molecules remain if the volume is changed to \(0.005 \ell ?\) What volume is occupied by 10,000 molecules at the initial temperature and pressure?

Short answer

The number of molecules remaining when the volume is changed to 0.005 L is approximately \(1.0 \times 10^{12}\) molecules. The volume occupied by 10,000 molecules at the initial temperature and pressure is \(5.0 \times 10^{-7} L\).

Step-by-step solution

Conversion to moles

Since the ideal gas law uses moles, we need to convert the number of molecules given (5.0 x 10^10) to moles by using Avogadro's number (6.022 x 10^23 molecules/mol): n_initial = (5.0 x 10^10 molecules) / (6.022 x 10^23 molecules/mol) = 8.303 x 10^{-14} mol

Applying Ideal Gas Law

Since the temperature and the number of moles are constant, we can equate their product with a constant, k: k = n_initial * R * T Since we are only concerned with the proportionality, we can ignore the R and T constants and just use: k = n_initial * V_initial = n_final * V_final where V_initial = 0.100 L and V_final = 0.005 L.

Finding remaining molecules

Using the proportionality equation, we can find the number of moles remaining: n_final = (n_initial * V_initial) / V_final = (8.303 x 10^{-14} mol)(0.100 L) / 0.005 L = 1.6606 x 10^{-12} mol Now, convert back to molecules using Avogadro's number: molecules remaining = n_final * Avogadro's number = (1.6606 x 10^{-12} mol)(6.022 x 10^23 molecules/mol) ≈ 1.0 x 10^12 molecules

Finding volume for 10,000 molecules

First, convert 10,000 molecules to moles: n = 10,000 molecules / (6.022 x 10^23 molecules/mol) = 1.661 x 10^{-19} mol Now, use the proportionality equation to find the volume occupied at the initial temperature and pressure: V = k / n = (8.303 x 10^{-14} mol * 0.100 L) / (1.661 x 10^{-19} mol) = 5.0 x 10^{-7} L This is the volume occupied by 10,000 molecules at the initial temperature and pressure.

Key concepts

Avogadro's Number

Avogadro's Number is crucial for converting between the microscopic scale of molecules and the macroscopic scale we can measure. It is defined as the number of constituent particles, usually atoms or molecules, that are contained in one mole of a substance. The value of Avogadro's Number is

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

This number acts as a bridge between the atomic world and real-world quantities. For instance, when calculating how many moles of a gas correspond to a certain number of molecules, we divide the number of molecules by Avogadro's Number.
In the original problem, Avogadro's Number is used to convert the given molecules into moles, a standard measurement useful for applying the Ideal Gas Law, which works in terms of moles rather than individual molecules.

Mole Conversion

Mole Conversion is a fundamental concept in chemistry that allows us to switch between different units of measurement like atoms, molecules, or liters, and moles. Understanding this process helps in accurately calculating quantities needed for chemical reactions or physical transformations.

• To convert molecules to moles, you divide by Avogadro's Number.
• To convert moles to molecules, you multiply by Avogadro's Number.

In the exercise provided, the initial number of molecules (\(5.0 \times 10^{10}\)) is converted to moles by dividing by Avogadro’s Number, resulting in the value \(8.303 \times 10^{-14}\) moles. Similarly, to revert to the number of molecules, you multiply the moles by Avogadro's Number. This conversion is key when using the Ideal Gas Law, which employs moles as a unit of measurement.

Volume and Pressure Relationship

The Volume and Pressure Relationship is a concept illustrated by Boyle's Law, which states that for a fixed amount of gas at constant temperature, the volume is inversely proportional to the pressure. While Boyle’s Law is not directly applicable here, a similar relationship is used, where volume changes affect the number of moles when pressure is constant.
In this problem, the volume change from \(0.100 \text{ L}\) to \(0.005 \text{ L}\) at constant temperature does not change the pressure but alters the number of molecules that fit into the space.

• The relation used is: \(n_{\text{initial}} \times V_{\text{initial}} = n_{\text{final}} \times V_{\text{final}}\).
• This implies if the volume decreases, the number of molecules per unit volume increases with constant pressure and temperature.

This relationship helps in determining how many molecules remain when the volume changes, using mole conversion results.

Gas Molecule Calculation

Gas Molecule Calculation involves using mathematical relationships to determine the amount of gas in terms of molecules or moles based on their environmental conditions, including pressure, volume, and temperature. This can often be calculated using the Ideal Gas Law or simple proportionality under constant conditions.
In the exercise, we calculated how many molecules remain when the volume decreases. After converting initial molecules to moles, we used the volume-mole relationship to determine final moles, then converted back to molecules to find the remaining count.

• Key calculation: First convert molecules to moles, apply the volume relationship, then convert back to molecules.

Similarly, for finding the volume of 10,000 molecules, we convert into moles, and apply the same relationship to find the occupied volume. This ensures we understand the space required for a specific count of gas molecules under given conditions.