The dimensions of a classroom are 4.20 m ×3.00 m ×2.50 m . (a) Find the number of molecules of a

step 1 All right, so I've got a classroom full of air. I have to find out this stuff. We're going to us

The dimensions of a classroom are 4.20 m ×3.00 m ×2.50 m ....

The dimensions of a classroom are $4.20 \mathrm{m} \times 3.00 \mathrm{m} \times$ 2.50 $\mathrm{m}$ . (a) Find the number of molecules of air in the classroom at atmospheric pressure and $20.0^{\circ} \mathrm{C}$ . (b) Find the mass of this air, assuming the air consists of diatomic molecules with molar mass $28.9 \mathrm{g} / \mathrm{mol} .$ (c) Find the average kinetic energy of the molecules. (d) Find the rms molecular speed. (e) What If? Assume the molar specific heat of the air is independent of temperature. Find the change in internal energy of the air in the room as the temperature is raised to $25.0^{\circ} \mathrm{C}$ . (f) Explain how you could convince a fellow student that your answer to part (e) is correct, even though it sounds surprising.

The dimensions of a classroom are $4.20 \mathrm{m} \times 3.00 \mathrm{m} \times$ 2.50 $\mathrm{m}$ . (a) Find the number of molecules of air in the classroom at atmospheric pressure and $20.0^{\circ} \mathrm{C}$ . (b) Find the mass of this air, assuming the air consists of diatomic molecules with molar mass $28.9 \mathrm{g} / \mathrm{mol} .$ (c) Find the average kinetic energy of the molecules. (d) Find the rms molecular speed. (e) What If? Assume the molar specific heat of the air is independent of temperature. Find the change in internal energy of the air in the room as the temperature
is raised to $25.0^{\circ} \mathrm{C}$ . (f) Explain how you could convince a fellow student that your answer to part (e) is correct, even though it sounds surprising.

Key Concepts

Temperature Dependence of Internal Energy

For ideal gases, the internal energy is solely a function of temperature. This means that any change in temperature directly alters the internal energy. Understanding this principle is crucial in assessing system energy changes and verifying unexpected results by linking experimental observations with theoretical predictions.

Root-Mean-Square (rms) Speed

The rms speed is a statistical measure of the speed of particles in a gas, derived from the kinetic molecular theory. It is calculated using the temperature and molar mass of the gas, and it provides insight into how the energy associated with particle motion translates into observable gas properties, such as pressure and temperature.

Internal Energy and Specific Heat

Internal energy in an ideal gas is the total kinetic energy of its molecules. The concept of specific heat, especially at constant volume, relates the change in the internal energy to a change in temperature. This relationship is key in determining how much energy is stored or released by the gas with a temperature change.

Avogadro's Number and Molar Relationships

Avogadro's number (6.022×10^23 molecules/mol) is a critical concept that connects the macroscopic world to the molecular scale by allowing one to convert between the number of moles and the number of individual molecules. This conversion is essential for problems that involve calculating the number of particles in a given volume of gas.

Ideal Gas Law

This law relates the pressure, volume, number of moles, and temperature of an ideal gas. It provides the foundational equation (PV = nRT) that is used to calculate quantities like the number of moles in a given volume under specified conditions, which is crucial when determining molecular counts and other thermodynamic properties.

Kinetic Molecular Theory

This theory explains the behavior of gases at the molecular level, asserting that gas molecules are in constant, random motion and that their thermal energy is directly proportional to the temperature. It forms the basis for calculating quantities such as the average kinetic energy of molecules and understanding how temperature influences molecular motion.

Transcript

00:01 All right, so i've got a classroom full of air.

00:06 I have to find out this stuff.

00:07 We're going to use pv equals nrt to figure it out.

00:12 First off, we're going to figure out how many moles there are of air.

00:16 The pressure is one atmosphere, which is 101 kilopascals.

00:24 The volume, length, times width, times height, is 31 .5 meters squared.

00:30 And is what we're looking for, r is a constant 8 .314 joules per mole per kelvin.

00:39 Temperature is 20 degrees celsius, which is 293 .15 kelvin, giving us 1 ,305 moles of air.

00:55 To figure out how many particles that is, we need to multiply by, we're going to take n and multiply it by 6 .022 times 10 to the 23, number of particles or atoms per mole, which gives us 7 .86 times 10 to the 26 atoms of air.

01:26 Each atom of air, nope, i'm sorry, it's per mole, is 28 .9 grams per mole.

01:36 So you multiply by the number of moles, and you will get the number, you will get the grams, which is 37 ,714, so 37 .7 kilograms.

01:52 Because having anything's in kilograms is most appropriate when trying to do other calculations.

02:00 Average kinetic energy, that is per atom, by the way, that will come in handy in a minute, is 3 over 2 times kb, which is 1 .38 times 10 to the negative 23, times the temperature, which is 293 .15, giving us 6 .07 times 10 to the negative 21 joules per atom.

02:34 So that's per atom, which is what they're asking for there.

02:41 But later on, they would like the total energy.

02:44 So if you take that and multiply it by the number of atoms there are, which is the 7 .86 times 10 to the 26, you will get, where to go, 4 .77 times 10 to the 6 joules of energy in the classroom per classroom.

03:18 Rms, the velocity root means squared, rms.

03:22 All right, so we take the square root of three times kb.

03:29 All right, 1 .38 times 10 to the negative 23 times the temperature and kelvin, of course, 293 .15 all over m sub 0 .0...

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