Question

You blow up a spherical balloon to a diameter of 50.0 cm until the absolute pressure inside is 1.25 atm and the temperature is 22.0\(^\circ\)C. Assume that all the gas is N\(_2\), of molar mass 28.0 g/mol. (a) Find the mass of a single N\(_2\) molecule. (b) How much translational kinetic energy does an average N\(_2\) molecule have? (c) How many N\(_2\) molecules are in this balloon? (d) What is the \(total\) translational kinetic energy of all the molecules in the balloon?

Short answer

(a) Mass of a single N₂ molecule is approximately 4.65 × 10⁻²³ g. (b) Average kinetic energy is approximately 6.1 × 10⁻²¹ J. (c) Number of molecules is approximately 2.1 × 10²³. (d) Total kinetic energy is approximately 1280 J.

Step-by-step solution

Convert Gas Temperature to Kelvin

To work with thermodynamic equations, the temperature must be in Kelvin. Convert from Celsius by adding 273.15.\[ T = 22.0 + 273.15 = 295.15 \text{ K} \]

Calculate Volume of the Balloon

The formula for the volume of a sphere is \( V = \frac{4}{3} \pi r^3 \). First, convert the diameter to radius in meters: \( r = \frac{50.0}{2} \text{ cm} = 0.25 \text{ m} \).\[ V = \frac{4}{3} \pi (0.25)^3 \approx 0.0654 \text{ m}^3 \]

Calculate Mass of Single N₂ Molecule

The molar mass of \(N_2\) is given as 28.0 g/mol. To find the mass of a single \(N_2\) molecule, use Avogadro's number, \(6.022 \times 10^{23}\) molecules/mol:\[ m_{\text{molecule}} = \frac{28.0 \text{ g/mol}}{6.022 \times 10^{23} \text{ molecules/mol}} \approx 4.65 \times 10^{-23} \text{ g} \]

Find Average Translational Kinetic Energy of an N₂ Molecule

The average translational kinetic energy of a molecule is given by \( \frac{3}{2}kT\), where \( k \) is the Boltzmann constant \(1.38 \times 10^{-23} \text{ J/K}\):\[ KE_{\text{avg}} = \frac{3}{2} \times 1.38 \times 10^{-23} \times 295.15 \approx 6.1 \times 10^{-21} \text{ J} \]

Calculate Number of N₂ Molecules in the Balloon

Using the Ideal Gas Law, \( PV = nRT \), where \(P\) is pressure, \(V\) is volume, and \(R = 8.314 \text{ J/(mol}\cdot\text{K)}\):Convert pressure to pascal: \(1.25 \text{ atm} = 1.25 \times 101325 \text{ Pa}\).\[ n = \frac{1.25 \times 101325 \times 0.0654}{8.314 \times 295.15} \approx 0.35 \text{ mol} \]Convert moles to molecules using Avogadro's number:\[ N = 0.35 \times 6.022 \times 10^{23} \approx 2.1 \times 10^{23} \text{ molecules} \]

Calculate Total Translational Kinetic Energy of All Molecules

Multiply the average kinetic energy per molecule by the total number of molecules to find the total kinetic energy:\[ KE_{\text{total}} = 6.1 \times 10^{-21} \times 2.1 \times 10^{23} \approx 1.28 \times 10^{3} \text{ J} \]

Key concepts

Kinetic Theory of Gases

The kinetic theory of gases is a fundamental concept that describes gases as a large number of small particles, which are in constant random motion. These particles collide elastically with each other and the walls of their container. This theory helps explain properties such as pressure, temperature, and volume in terms of molecular behavior and energy.
Gases have molecules that move freely and independently, contributing to the pressure when they hit the walls of their container. The assumptions made by the kinetic theory include:

• Gas molecules are in continuous, random motion.
• All collisions between molecules or with walls are perfectly elastic, meaning no net loss in kinetic energy occurs.
• The volume of the gas molecules is negligible compared to the volume of the container.
• There are no intermolecular forces apart from during collisions.

Understanding this theory is crucial for using the Ideal Gas Law, which describes the state of a gas under specified conditions of pressure, volume, and temperature.

Molecular Mass Calculation

The molecular mass of a gas involves understanding the total mass of all atoms within a single molecule of the gas. In calculations, we often express it in grams per mole (g/mol). For example, nitrogen gas, which is given as the gas in the balloon, has a molar mass of 28.0 g/mol. This value represents the mass of one mole of nitrogen molecules and is vital for further calculations.
To find the mass of an individual nitrogen molecule, we need to use Avogadro's number. The formula applied here is:

• Mass of one molecule = \( \frac{\text{Molar Mass}}{\text{Avogadro's Number}} \)

Given that Avogadro's number is \(6.022 \times 10^{23}\), we can accurately determine the mass of a single gas molecule by dividing the molar mass by this constant.

Translational Kinetic Energy

Translational kinetic energy corresponds to the energy due to the motion of molecules. At a certain temperature, molecules possess a specific amount of kinetic energy which can be calculated. The formula used for this is the kinetic theory expression:

\[ KE_{\text{avg}} = \frac{3}{2} k T \]
Here, \( k \) is the Boltzmann constant (\(1.38 \times 10^{-23} \text{ J/K}\)) and \( T \) is the temperature in Kelvin. Using this equation, you can determine the average kinetic energy per molecule.
When temperature increases, so does the average kinetic energy, as molecules move more rapidly. This energy remains essential to understanding the molecular behavior and how they disperse energy among themselves in a gas.

Avogadro's Number

Avogadro's number \( (6.022 \times 10^{23} \text{ molecules/mol}) \) plays a pivotal role in chemistry and physics. It defines the number of molecules or atoms in one mole of a substance, offering a bridge between macroscopic and microscopic properties of matter.
It aids in converting the amount of substance (in moles) into specific counts of molecules, allowing for intricate calculations like determining the total number of molecules in a defined volume of gas.
This conversion is vital for tasks such as calculating the number of molecules in a balloon filled with nitrogen or finding out the total mass of substances at the molecular level by using the molar mass. By knowing this fundamental constant, we can transition from the macroscale measurements to the atomic scale calculations effectively.

Boltzmann Constant

The Boltzmann constant \( (k = 1.38 \times 10^{-23} \text{ J/K}) \) is an essential physical constant that links the macroscopic properties of gases to their microscopic movements and energy levels.
Named after Ludwig Boltzmann, it serves in translating the kinetic energy of molecules at a microscopic level to observable macroscopic quantities like temperature. It plays an integral role in formulas related to kinetic theory, particularly in calculating the average translational kinetic energy of a molecule.
This constant ensures that temperature is expressed as a measure of the average kinetic energy of particles in a substance. Understanding the Boltzmann constant is crucial to working with equations describing gas behaviors, as it provides the means to combine thermal dynamics with atomic-scale physics seamlessly.