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

Answer: There are approximately \(8.47\times10^{21}\) helium atoms inside the balloon. Their average kinetic energy is approximately \(6.07\times10^{-21}\,\mathrm{J}\). The root-mean-square speed of the atoms is approximately \(1371.68\,\mathrm{m/s}\).

Step-by-step solution

Calculate the volume of the balloon using diameter

To calculate the volume of the balloon, we can use the formula for the volume of a sphere: \(V = \frac{4}{3} \pi r^3\) where \(V\) is the volume of the sphere, and \(r\) is the radius. Since the diameter of the balloon is given, we can calculate the radius as half of the diameter: \(r = \frac{40}{2}=20 \mathrm{~cm}\) Now plug in the value of \(r\) into the volume formula: \(V = \frac{4}{3} \pi (20)^3\) Calculating the volume, we get: \(V \approx 33510.3 \mathrm{~cm^3}\)

Convert the volume to SI units and temperature to Kelvin

To work with the ideal gas law, we need to have the volume in SI units (cubic meters) and the temperature in Kelvin. To do this, we can convert from centimeters to meters, and from celsius to Kelvin: \(V = 33510.3 \mathrm{~cm^3}\times\frac{1\,\mathrm{m^{3}}}{1000000\,\mathrm{cm^{3}}} \approx 3.35103\times10^{-2}\,\mathrm{m^3}\) \(T = 20.0^{\circ} \mathrm{C} + 273.15 = 293.15 \,\mathrm{K}\)

Use the ideal gas law to find the number of moles of helium

Now we can use the ideal gas law to find the number of moles (n) of helium in the balloon: \(PV = nRT\) where \(P\) is the pressure (in atm), \(V\) is the volume (in meters), \(n\) is the number of moles, \(R\) is the ideal gas constant (0.08206 L·atm/mol·K), and \(T\) is the temperature (in Kelvin). We are given \(P = 1 \,\mathrm{atm}\), and we have found \(V\) and \(T\). Now we can rearrange the equation to solve for \(n\) and convert volume to liters: \(n = \frac{PV}{RT}\) \(n = \frac{1 \,\mathrm{atm} \times (3.35103 \times 10^{-2} \,\mathrm{m^3} * \frac{1000\,\mathrm{L}}{1\,\mathrm{m^{3}}})}{0.08206 \mathrm{L\cdot atm/mol\cdot K} \times 293.15 \,\mathrm{K}}\) Calculating the number of moles, we get: \(n \approx 0.0141 \,\mathrm{mol}\)

Use Avogadro's number to find the number of helium atoms

Now that we have the number of moles, we can use Avogadro's number to find the number of helium atoms inside the balloon: \(N = n \times N_A\) where \(N\) is the number of helium atoms, \(n\) is the number of moles, and \(N_A\) is Avogadro's number (\(6.022\times 10^{23} \,\mathrm{atoms/mol}\)): \(N = 0.0141 \,\mathrm{mol} \times 6.022 \times 10^{23} \,\mathrm{atoms/mol}\) Calculating the number of helium atoms, we get: \(N \approx 8.47\times 10^{21} \,\mathrm{atoms}\)

Calculate average kinetic energy per atom

The average kinetic energy per atom can be found using the formula: \(KE_{avg} = \frac{3}{2} kT\) where \(KE_{avg}\) is the average kinetic energy per atom, \(k\) is Boltzmann's constant (\(1.381\times10^{-23}\,\mathrm{J/K}\)), and \(T\) is the temperature (in Kelvin): \(KE_{avg} = \frac{3}{2} \times 1.381 \times 10^{-23} \,\mathrm{J/K} \times 293.15 \,\mathrm{K}\) Calculating the average kinetic energy per atom, we get: \(KE_{avg} \approx 6.07 \times 10^{-21} \,\mathrm{J}\)

Calculate the root-mean-square speed of helium atoms

Finally, we can find the root-mean-square speed for helium atoms using the formula: \(v_{rms} = \sqrt{\frac{3kT}{m}}\) where \(v_{rms}\) is the root-mean-square speed, \(k\) is Boltzmann's constant, \(T\) is the temperature (in Kelvin), and \(m\) is the mass of a single helium atom. To find the mass of a helium atom, we can use the molar mass of helium (\(4.00\,\mathrm{g/mol}\)) and Avogadro's number: \(m = \frac{4.00\,\mathrm{g/mol}}{6.022\times 10^{23}\,\mathrm{atoms/mol}} = 6.64\times 10^{-24}\,\mathrm{g}\) Now, convert the mass of a helium atom to kilograms: \(m = 6.64\times 10^{-24}\,\mathrm{g} \times \frac{1\,\mathrm{kg}}{1000\,\mathrm{g}} = 6.64\times 10^{-27}\,\mathrm{kg}\) Plug the values into the equation for root-mean-square speed: \(v_{rms} = \sqrt{\frac{3\times1.381\times 10^{-23}\,\mathrm{J/K} \times 293.15\,\mathrm{K}}{6.64\times 10^{-27}\,\mathrm{kg}}}\) Calculating the root-mean-square speed, we get: \(v_{rms} \approx 1371.68\,\mathrm{m/s}\) So, the answers are: a) There are approximately \(8.47\times10^{21}\) helium atoms inside the balloon. b) The average kinetic energy of the atoms is approximately \(6.07\times10^{-21}\,\mathrm{J}\). c) The root-mean-square speed of the atoms is approximately \(1371.68\,\mathrm{m/s}\).

Key concepts

Kinetic Theory and Gases

Understanding the kinetic theory is essential to grasp the behavior of gases. It explains how particles in a gas move and behave. According to this theory, gas atoms and molecules are in constant motion. They move in straight lines until they collide with either another particle or the walls of their container.

In a sample container such as a helium balloon, the pressure is created due to these collisions. Temperature directly relates to how fast these molecules are moving. The higher the temperature, the faster the molecules move. If we imagine a balloon where the helium atoms are moving around energetically, this movement creates force against the balloon's walls. A key takeaway is that gas characteristics like temperature and volume significantly influence particle speed and energy.

• Gases consist of tiny, fast-moving particles.
• Pressure results from collisions within the container.
• Temperature is a direct measure of the average kinetic energy of particles.

Root-Mean-Square Speed

In the context of kinetic theory, the speed of gas particles is a critical aspect. The root-mean-square (RMS) speed helps measure the effective speed of particles in a gas sample. Rather than calculating the average or mean speed, RMS considers the square root of the average of the squares of the particle speeds, offering insights into the energy they possess.

To find the RMS speed, we use the equation:\[ v_{rms} = \sqrt{\frac{3kT}{m}} \]Here, \(k\) is Boltzmann's constant, \(T\) is the absolute temperature, and \(m\) is the mass of a single particle. This equation is vital to calculate how fast, on average, the particles in a gas like helium would be moving. The RMS speed is a key parameter for understanding how gas behaves under certain conditions.

• RMS speed reflects average energy in motion of particles.
• Higher temperatures increase RMS speed.
• Lighter particles move faster at the same temperature.

Avogadro's Number

Avogadro's number is a cornerstone concept in chemistry, especially when dealing with gases. This number, \(6.022 \times 10^{23}\), defines how many particles—such as atoms or molecules—are present in a mole. It provides a bridge between the macroscopic and microscopic worlds.

For example, with gases like helium, knowing the number of moles helps determine the exact number of atoms in a given volume of gas. When the ideal gas law, \( PV = nRT \), is utilized to find the moles, multiplying by Avogadro's number converts this to a tangible count of atoms. In the helium balloon problem, by using this number alongside the calculated moles, it becomes straightforward to determine there are approximately \(8.47 \times 10^{21}\) helium atoms in the balloon.

• Essential for converting moles to number of particles.
• Makes microscopically small numbers tangible.
• Used extensively in chemistry and physics to understand quantity at the atomic level.