You blow up a spherical balloon to a diameter of 50.0 cm until the absolute pressure inside is 1

step 1 Hi everyone here it is given. A radius of Valoon is 50 cm at pressure 1 .25 atmosphere and tempe

You blow up a spherical balloon to a diameter of 50.0 cm...

You blow up a spherical balloon to a diameter of 50.0 $\mathrm{cm}$ until the absolute pressure inside is 1.25 atm and the temperature is $22.0^{\circ} \mathrm{C}$ . Assume that all the gas in $\mathrm{N}_{2}$ is of molar mass 28.0 $\mathrm{g} / \mathrm{mol}$ . (a) Find the mass of a single $\mathrm{N}_{2}$ molecule. (b) How much translational kinetic energy does an average $\mathrm{N}_{2}$ molecule have? (c) How many $\mathrm{N}_{2}$ molecules are in this balloon? (d) What is the total translational kinetic energy of all the inolecules in the balloon?

Key Concepts

Total Kinetic Energy of a Gas

The total translational kinetic energy of a gas is found by multiplying the average kinetic energy per molecule by the total number of molecules. This concept combines microscopic kinetic theory with macroscopic measurements, enabling the conversion of molecular-level energy descriptions to a total energy content for the whole system. It illustrates the additive nature of energy in a gas comprising many individual particles.

Spherical Geometry and Volume Calculation

For problems involving a spherical container, calculating the volume is key. The volume of a sphere is given by (4/3)?r^3, where r is the radius of the sphere. This concept is important when the gas is confined in a spherical container, as it provides the spatial context needed for applying gas laws and determining the amount of substance present.

Ideal Gas Law

The ideal gas law (PV = nRT) relates the pressure, volume, temperature, and amount of gas (in moles) in a system. This equation is used to determine the number of moles of gas present when the pressure, volume, and temperature are known. Once the number of moles is calculated, it can be converted to the number of molecules using Avogadro’s number, making it a critical tool for linking macroscopic gas properties to microscopic particle counts.

Molar Mass and Avogadro’s Number

This key concept involves converting the molar mass of a substance into the mass of a single molecule using Avogadro’s number. Since the molar mass represents the mass per mole (with one mole containing approximately 6.022×10^23 molecules), dividing the molar mass by Avogadro’s number yields the mass of an individual molecule. This is a fundamental conversion in chemistry that bridges macroscopic measurements with microscopic particles.

Equipartition Theorem for Translational Kinetic Energy

The equipartition theorem states that, at thermal equilibrium, the energy associated with each quadratic degree of freedom is (1/2)kT, where k is the Boltzmann constant and T is the absolute temperature. For a molecule moving in three dimensions, the average translational kinetic energy is therefore (3/2)kT. This principle is essential for relating the temperature of a gas to the kinetic energy of its molecules.

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