Question

At Party City, you purchase a helium-filled balloon with a diameter of \(40.0 \mathrm{~cm}\) at \(20.0^{\circ} \mathrm{C}\) and at \(1.00 \mathrm{~atm} .\). a) How many helium atoms are inside the balloon? b) What is the average kinetic energy of the atoms? c) What is the root-mean-square speed of the atoms?

Short answer

Question: Determine the following properties of a helium-filled balloon with a diameter of 40 cm, temperature of 20.0°C, and pressure of 1.00 atm: a) the number of helium atoms inside the balloon, b) total average kinetic energy of helium atoms in the balloon, and c) root-mean-square speed of helium atoms. Answer: The step-by-step solution is provided above. To summarize, apply the Ideal Gas Law to find the number of moles of helium in the balloon and then calculate the number of helium atoms. Next, calculate the average kinetic energy per molecule and the total average kinetic energy using the given temperature and Boltzmann's constant. Lastly, use the molar mass of helium and temperature to find the root-mean-square speed of the helium atoms.

Step-by-step solution

Calculate the volume of the balloon

Given the diameter of the balloon to be 40 cm, we can compute its volume using the formula for the volume of a sphere: \(V=\frac{4}{3}\pi r^{3}\), where r is the radius. Since the diameter of the sphere is 40 cm, the radius is half of it (20 cm). So, we have: \(V=\frac{4}{3}\pi (20\,\mathrm{cm})^{3}\).

Convert volume to liters

To use the Ideal Gas Law, we need to have the volume in liters. We can use the relationship: 1 L = 1000 cm³. \(V=\frac{4}{3}\pi (20\,\mathrm{cm})^{3} \times \frac{1\,\mathrm{L}}{1000\,\mathrm{cm}^3}\).

Apply the Ideal Gas Law to find moles of helium

The Ideal Gas Law is: \(PV=nRT\), where P is pressure, V is volume, n is the number of moles, R is the Ideal Gas Constant, and T is temperature in Kelvin. We are given P = 1 atm, we have calculated V in Step 2, and we know R = 0.0821 L.atm/mol.K (use this value for R in the problem). We need to convert the given temperature in Celsius to Kelvin by adding 273.15: T = 20.0°C + 273.15. Now, we can find the number of moles (n) of helium in the balloon: \(1.00\,\mathrm{atm} \times V = n \times 0.0821\,\mathrm{L\cdot atm\cdot mol^{-1}\cdot K^{-1}} \times (20.0+273.15)\,\mathrm{K}\)

Calculate the number of helium atoms

Now that we have the number of moles (n) of helium, we can find the number of helium atoms using Avogadro's number (6.022 x 10²³ atoms/mol): Number of helium atoms = n x Avogadro's number. #b) Average kinetic energy#

Calculate the average kinetic energy per molecule

We can use the equation for the average kinetic energy of a gas particle in terms of temperature: Average kinetic energy per molecule = \(\frac{3}{2}kT\), where k is the Boltzmann's constant (1.38 x 10⁻²³ J/K) and T is the temperature in Kelvin (already calculated in part a).

Calculate the total average kinetic energy

Since we know the number of helium atoms (calculated in part a), we can multiply it by the average kinetic energy per molecule to get the total average kinetic energy inside the balloon: Total average kinetic energy = Number of helium atoms x Average kinetic energy per molecule. #c) Root-mean-square speed#

Calculate the molar mass of helium

The molar mass of helium is 4.00 g/mol. To use it in the root-mean-square speed equation, we need to convert it to kg/mol: Molar mass of helium = \(\frac{4.00\,\mathrm{g/mol}}{1000\,\mathrm{g/kg}}\)

Calculate the root-mean-square speed of the atoms

We can use the equation for the root-mean-square speed of a gas particle: Root-mean-square speed = \(\sqrt{\frac{3RT}{M}}\), where R is the Ideal Gas Constant (8.314 J/mol.K), T is the temperature in Kelvin (already calculated in part a), and M is the molar mass of helium in kg/mol (calculated in Step 1). Solve the equation to find the root-mean-square speed.