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

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

$\bullet$$\bullet$ 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 is $\mathrm{N}_{2},$ 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 molecules in the balloon?

$\bullet$$\bullet$ 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 is $\mathrm{N}_{2},$ 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 molecules in the balloon?

Key Concepts

Ideal Gas Law

The ideal gas law (PV = nRT) connects the macroscopic properties of a gas—pressure, volume, and temperature—to the amount of gas in moles. It is fundamental in thermodynamics for determining the number of moles present in a given volume or under given conditions, thus linking the microscopic behavior of gas molecules to observable quantities.

Molecular Mass and Avogadro's Number

Molecular mass provides a means to relate the mass of a mole of a substance to the mass of a single molecule by using Avogadro's number. Dividing the molar mass by Avogadro's number converts macroscopic mass units into the mass of an individual molecule, bridging the gap between the molecular and bulk scales.

Equipartition Theorem and Translational Kinetic Energy

The equipartition theorem states that energy is equally distributed among all available degrees of freedom. For translational motion in three dimensions, each molecule possesses an average kinetic energy of (3/2)kT, where k represents Boltzmann's constant and T is the absolute temperature. This concept is essential in understanding how temperature correlates with the average kinetic energy of gas molecules.

Volume Calculation for Spherical Objects

Calculating the volume of a spherical container is achieved with the formula (4/3)?r³. This geometric relation is crucial when determining the space available for gas molecules in problems involving spherical balloons or other spherical vessels.

Temperature Unit Conversion

Thermodynamic equations require absolute temperature measurements in Kelvin. Converting temperatures from Celsius to Kelvin by adding 273.15 is necessary to ensure consistency and accuracy when applying formulas such as the ideal gas law and when calculating kinetic energies.

Transcript

00:01 Diameter of inflated balloon is 50 cm so here we have diameter is equal to 50 centimeter so the radius of balloon is equal to 25 centimeter which is equal to 25 multiply 10 to the power minus 2 meter and the pressure inside the balloon is 1 .5 25 atmosphere and the temperature upgas inside the balloon is 22 degree celsius which is equal to 29 .15 kelvin so now in the first part of this problem we want to calculate mass of a single nitrogen molecule mass of a single nitrogen molecule mass of nitrogen molecule the gas inside the balloon is nitrogen and molar mass of nitrogen is equal to 28 ground per more as we have the relation for molar mass that molar mass is equal to mass of a single molecule multiply a regardless number so from here we have mass of single molecule is equal to molar mass upon a regardless number here molar mass of the gases 28 gram per mole and we have a regardless number that is 6 .0 2 to multiply 10 to the power 23 molecules per mall so we get mass of a single molecule is equal to 4 .65 multiply 10 to the power minus 23 ground per mole so we have mass of a single molecule is equal to 4 .65 multiply 10 to the power minus 26 kilograms per mole now in the second part of this problem we want to calculate the translational kinetic energy of each molecule as translational kinetic energy is equal to half of mass times velocity of a single molecule here's velocity of single molecule is v rms and vrrm s is equal to square root of 3k t upon m so we get vrms secure is equal to 3k t upon m now by putting the value of we rms we get trostnational kinetic energy is equal to half time times m into 3k t upon m here m and m will be cancelled so we get transnational kinetic energy is equal to 3 k t upon now by putting the values here 3 multiply the value of k that is the bolzman's constant is 1.

05:33 3 at 1 multiply 10 to the power minus 23 jol per molecule per kelvin and temperature here is 295 .15 kelvin upon 2...

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