Assume a hydrogen atom is a sphere with diameter 0.100 nm and a hydrogen molecule consists of tw

step 1 Okay, so for this problem, I'm going to use the six steps for Part A. Step number one is what is

Assume a hydrogen atom is a sphere with diameter 0.100 nm...

Key Concepts

Ideal Gas Law

The Ideal Gas Law relates pressure, volume, temperature, and the number of particles in a gas. It provides the conceptual framework for determining how many molecules are present in a given volume of gas under specified conditions, which is essential for calculating the physical space occupied by the gas particles.

Sphere Volume Calculation

This concept involves using the mathematical formula for the volume of a sphere, V = (4/3)?r³, to compute the space occupied by spherical objects. It is a fundamental tool for estimating the volume of atoms or molecules when they are approximated as spheres in various physical and chemical contexts.

Volume Fraction

The volume fraction is the ratio of the volume occupied by a component (such as a molecule or nucleus) to the total volume of a system or container. This concept is critical for understanding how much of a given space is physically taken up by the constituent objects, as opposed to being empty space.

Scale Comparison in Physics

Scale comparison involves studying how different physical dimensions vary by many orders of magnitude, such as comparing the size of an atom to the size of its nucleus. This is essential for understanding the relative sizes and distributions of different components within a system, highlighting the vast differences between atomic and subatomic scales.

Transcript

00:01 Okay, so for this problem, i'm going to use the six steps for part a.

00:09 Step number one is what is the question asking you for? it's asking for the fraction of space, these hydrogen molecules occupy within a tank.

00:19 So it's not exactly a variable, so i'm going to use jump to step number two.

00:27 See this as a drawing first.

00:28 So we have all of these hydrogen molecules inside of a tank.

00:34 And we want to know what fraction of space that's occupied by the hydrogen molecules.

00:40 So on the top of this fraction, we want to know the volume of all of the hydrogen molecules.

00:47 So that'll be the number of hydrogen molecules.

00:52 I'm going to call this nh2 times the volume of one hydrogen molecule.

01:01 So v volume of hydrogen molecule.

01:04 Hydrogen.

01:05 And this is all going to be over the entire volume of the tank.

01:12 So that's what we're looking for that fraction here.

01:16 So there's multiple variables in one.

01:18 What we're looking for to find this fraction.

01:20 And then step three, what does the problem give you in terms of variables? over here we have the diameter of one hydrogen atom, which is 0 .1 nanometers.

01:43 We have the temperature of zero to degrees celsius and we have a pressure 1 atm.

02:06 So now we wanna see what equations we can apply.

02:10 And step four, so as we see the pressure, temperature, and volume, i'm sure you can assume that we can use the ideal gas law, which is pv equals nrt.

02:33 Now here we have a pressure.

02:36 The volume is in our unknown.

02:41 Number of moles we need to use some more thinking of how we're going to get that variable.

02:46 This r boltzmann's constant, which is r equals 0 .0821 liters times atm over moles times kelvin.

03:15 So i like to keep the units constant here.

03:18 So i want to change just liters into the metric system.

03:22 So that would be meters cubed.

03:26 So i'm going to multiply this in one liter.

03:29 There is 0 .001 meters cubed and all i'm doing here is changing the unit.

03:38 So that leads me with 8 .21 times 10 to the negative 5.

03:46 Now i have a meters cubed on top times atm over mole times kelvin.

04:01 All right.

04:02 So coming back to the pv equals an it equation, we have a temperature here.

04:07 So now that moles we've got to figure out.

04:13 And up here, we've got to find out the number of hydrogen molecules and the volume of a hydrogen molecule.

04:22 But the volume of a hydrogen molecule we can find from the diameter up here.

04:32 So this would be.

04:41 So the volume of a sphere is four thirds pi, r cubed.

04:49 Since this is a diameter, we've got to use half of it.

04:58 That's going to be cubed.

05:00 I remember there's two of them, so we've got to multiply this by two.

05:04 So that'll be the volume of a hydrogen molecule.

05:09 So that checks out there.

05:12 And as we said, the volume is in the ideal gas law equation.

05:17 So now we have to find what this, this is number of moles of a h2 molecule.

05:24 I'll just add this little h2 on the bottom right here.

05:30 And the number of h2 molecules.

05:35 So what variable or equation do we know that relates the number of moles and the number of atoms, since that's what we're looking for here in both of these variables? so we know that avogadjo's number is a number of atoms per mole.

05:56 So if i write avogadjo's number, which is equal to 6 .022, atoms per mole.

06:21 So we know that the units for the number of hydrogen molecules would be in atoms and moles units and n is a number of moles so moles of h2.

06:48 So that's a way that we can relate to of 6 .022 times 10 to the 23rd.

07:23 So now i'm gonna rewrite this in terms of moles of h2, you simply re -arraging the equation.

07:36 So the number of moles at h2 will be the number of h2 atoms over avogadjo's number.

07:55 So now i can plug in the number of moles back into the ideal gas law down here.

08:03 So pv equals number of h2 over avagadjo's number times bothman's constant times t.

08:28 So we could even move this p to the bottom right here, this pressure, to get a volume by itself...

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