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

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

Assume a hydrogen atom is a sphere with diameter $0.100 \mathrm{nm}$ and a hydrogen molecule consists of two such spheres in contact. (a) What fraction of the space in a tank of hydrogen gas at $0^{\circ} \mathrm{C}$ and $1.00 \mathrm{~atm}$ is occupied by the hydrogen molecules themselves? (b) What fraction of the space within one hydrogen atom is occupied by its nucleus, of radius $1.20 \mathrm{fm} ?$

Assume a hydrogen atom is a sphere with diameter $0.100 \mathrm{nm}$ and a hydrogen molecule consists of two such spheres in contact.
(a) What fraction of the space in a tank of hydrogen gas at $0^{\circ} \mathrm{C}$ and $1.00 \mathrm{~atm}$ is occupied by the hydrogen molecules themselves?
(b) What fraction of the space within one hydrogen atom is occupied by its nucleus, of radius $1.20 \mathrm{fm} ?$

Key Concepts

Atomic and Nuclear Dimensions

Understanding the significant difference between the scales of atomic and nuclear dimensions is essential. While the hydrogen atom has dimensions on the order of tenths of a nanometer, its nucleus is much smaller, on the order of femtometers. This disparity explains why the nucleus occupies only a minute fraction of the atomic volume, which is fundamental in problems evaluating spatial occupancy at different scales.

Hard Sphere Model

The hard sphere model approximates atoms or molecules as rigid, non-deformable spheres that occupy space but do not interact with each other except through elastic collisions. This concept is key in understanding the fraction of the total volume that is actually occupied by the discrete particles, such as in computing the volume fraction in a gas or inside an individual atom.

Ideal Gas Law

The ideal gas law, expressed as PV = nRT, is vital for relating pressure, volume, and temperature of a gas. In problems concerning the space occupied by gas molecules under certain conditions, this law is used to determine the number of molecules per unit volume or the volume available to the gas molecules before considering their finite size.

Volume of a Sphere

The formula for the volume of a sphere, (4/3)?r³, is crucial for calculating the physical space taken up by spherical objects like atoms or nuclei. This geometric concept allows one to determine the volume of a hydrogen atom assumed to be a sphere, as well as the volume of the nucleus, thereby enabling the computation of fractional occupancies in a given volume.

Transcript

00:01 For this question, we are looking at these three different elements that have the same nucleon number.

00:09 They have the same total number of nucleons, they have the same mass number.

00:12 Basically, they are called isobars.

00:18 Now, because they have different elements, they have different properties, of course.

00:24 So the first one we want to find is, we want to identify which of them has the highest neutron to proton ratio.

00:31 So basically neutron divided by protons must be the highest.

00:35 So the largest number of protons, sorry, the largest number of neutrons, which basically means we have the lowest number of protons.

00:44 And that's very simple because cesium over here only has 55 protons.

00:50 Therefore we expect ccium to have the greatest neutron to proton ratio.

01:05 Next, which one of them do you expect to have the highest binding energy per nucleon? well, to have a high binding energy per nucleon, it means that your nuclei is very stable.

01:22 Most likely it will not decay, or it will take a very long time to decay.

01:30 And right from our question itself, really given that these two elements, right, the c -3 and the other pr element actually decays very by different means, right? it's radioactive and they decay, whereas our lengthenum actually doesn't decay, right? it's a stable nuclei.

01:58 So we expect actually the lanternum to be the most stable, and therefore it is highest binding energy...

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