Question

If \(5.55 \times 10^{22}\) atoms of He occupy a volume of \(2.06 \mathrm{~L}\) at \(0^{\circ} \mathrm{C}\) at 1.00 atm pressure, what volume do \(2.08 \times 10^{23}\) atoms of He occupy under the same conditions?

Short answer

The volume is 7.72 L.

Step-by-step solution

Understand the problem

We are given a certain volume occupied by a specific number of helium atoms under certain conditions of temperature and pressure. We need to find the volume occupied by a different number of helium atoms under the same conditions.

Use Avogadro's Law

Avogadro's Law states that equal volumes of gases, at the same temperature and pressure, will contain the same number of molecules or atoms. Hence, the volume is directly proportional to the number of atoms or molecules: \( V_1/V_2 = N_1/N_2 \).

Set up the equation

Let \( V_1 = 2.06 \) L, \( N_1 = 5.55 \times 10^{22} \), and \( N_2 = 2.08 \times 10^{23} \). We need to find \( V_2 \). Set up the equation as follows: \( \frac{V_1}{V_2} = \frac{N_1}{N_2} \).

Solve for V2

Rearrange the equation to solve for \( V_2 \): \( V_2 = \frac{V_1 \times N_2}{N_1} \). Plug in the values: \( V_2 = \frac{2.06 \times 2.08 \times 10^{23}}{5.55 \times 10^{22}} \).

Calculate

Calculate the expression to find \( V_2 \): \( V_2 = \frac{2.06 \times 2.08}{5.55} \times 10 = 7.72 \text{ L} \).

Key concepts

Gas Volume Calculation

In this exercise, we want to determine the volume that a certain quantity of helium atoms will occupy under specified conditions. To solve this, we employ a key principle from chemistry known as Avogadro's Law. Avogadro's Law helps us predict the relationship between the volume of a gas and the number of atoms or molecules it contains. The mathematical expression of this law is: \[ V_1 / V_2 = N_1 / N_2 \] where:

• \( V_1 \) and \( V_2 \) are the initial and final volumes of the gas.
• \( N_1 \) and \( N_2 \) are the initial and final numbers of atoms respectively.

This equation implies that the ratio of the two volumes is equal to the ratio of the two amounts of substance. By rearranging the formula, we can solve for the unknown volume. This approach is critical when considering changes in the amount of gas while maintaining consistent conditions of temperature and pressure.

Temperature and Pressure Conditions

The conditions of temperature and pressure are crucial when dealing with gases. For this problem, conditions are set at 0°C and 1.00 atm. These are standard conditions often used in chemistry to ensure clarity and consistency across calculations.

• Temperature (0°C): At 0°C, the behavior of gases is quite predictable, which simplifies calculations. Remember, 0°C can also be converted to 273.15 K if needed.
• Pressure (1 atm): The pressure is kept at one atmosphere, which is roughly the average atmospheric pressure at sea level.

These conditions allow us to use Avogadro's Law effectively, as they ensure that any changes in volume are solely due to the changes in the number of helium atoms. Keeping temperature and pressure constant enables a direct application of the law since these factors can otherwise influence the gas volume significantly.

Helium Atoms and Volume

Helium, a noble gas, is known for its simplicity and inertness, making it an ideal candidate for studying basic gas laws. In this scenario, we started with \(5.55 \times 10^{22}\) helium atoms occupying a volume of 2.06 liters. Then, we examined what happens when we increase the number of atoms to \(2.08 \times 10^{23}\). Based on Avogadro's Law, the volume of a gas should increase proportionally with the number of atoms if pressure and temperature remain constant.To calculate the new volume, we plug the values into our formula: \[ V_2 = \frac{2.06 \times 2.08 \times 10^{23}}{5.55 \times 10^{22}} \]Carrying out the calculations gives us \( V_2 = 7.72 \text{ L} \). This calculation confirms that by increasing the number of helium atoms while maintaining the pressure and temperature, the volume of the gas increases proportionally, as expected according to Avogadro's Law.