Is the boundary of an atom well defined?

Question

That question might seem weird, but that's what I have been imagining lately; going through Bohr's theory on the hydrogen atom. He clearly defines the radius of an orbit with an exact mathematical expression. But what do they mean when they say any atom (say $\ce{Na}$) has a radius $r$. Are we to consider atoms as solid, rigid spheres? If that's the case, what about water?

If all the atoms are defined as solid, rigid, spherical balls, how come an aggregation of so many solid objects be a liquid? How can it flow? So, in a nutshell what is the boundary of an atom made up of, or even this: does it have a defined boundary?

This troubles me a lot. I think maybe the modern atomic model has dealt with this problem. I haven't read it in detail, and I would be highly thankful to any references.

Answer 1

No, the boundary of an atom is not well defined. According Quantum mechanics the electron density around an atoms decreases exponentially, thus getting asymptotically close to zero.

Considering atoms as rigid spheres is only an approximation. The criteria used to decide on the actual value of the atomic radius is arbitrary. You can for example choose the expectation value $\langle\hat r\rangle$ of the electronic wavefunction. For the Hydrogren atom, this results in the bohr radius $a_0$. But you might as well choose the maximum of the electron density ($\frac{3}{2}a_0$ for $\ce{H}$), or something based on experiments.

Now the aggregate state is something completely different and not directly related to atomic radius and whether they are rigid sphere or not. Solid means the molecules/atoms of a substance are fixed in their positions. They can vibrate, but do not change their average position with respect to each other. In gases molecules may move freely. And a liquid is something in between, you have some order if you look at a small group of molecules, but not on a larger scale as for solids.

Answer 2

As was already mentioned, the boundary of an atom is everything but well-defined. Quantum mechanics tell us this very well: the probability of finding an electron at the distance $r$ from the nucleus is proportional to $\mathrm e^{-r}$, meaning that it will never hit zero.^[1] The lack of a zero-point means that there is no cutoff after which we can clearly say the atom is over. Instead, when performing quantum calculations an arbitrary cutoff value is chosen. The boundary within which the probability of finding the electron is larger than the cutoff value is then plotted as the ‘extent of the orbital’.

However, even if atoms were rigid spheres that does not mean your macroscopic postulate (that rigid spheres cannot form a liquid) holds true because you are misestimating the difference between microscopic and macroscopic properties. I present you the following example:

We will both agree that sand is made up of rigid solid particles even if they are not perfect spheres. Yet, sand shows a few properties that almost make it look liquid if you look from far away. For example, it can ‘flow down’ a slope much in the same way as water could. And also if you try to pile it into a pyramid, it will flow outwards to a certain extent as water (or better: mercury) drops would. You might extend this analogy to marbles which are clearly spherical (usually). You cannot stack them on a blank surface as they will roll away much like you cannot stack water molecules because they will flow to the side.

The main difference between my macroscopic examples and actual molecules is the type and extent of attractive forces between them. While the rough texture of sand can be said to give it some kind of a surface tension (like water and mercury), the attractive forces between the particles are still low and the main force acting on them is gravity (of the Earth, not of other grains of sand). The same thing is true for the marbles example which even lack surface tension and are only attracted to the Earth by gravity. Actual molecules have much stronger intermolecular forces that dominate over the external gravity force source in the microscopic world. That contributes to the final edge of a difference between many grains of sand/many marbles and a sample of water.

All I have discussed above is, however, completely independent of the question whether the spheres that make up atoms are rigid and solid or not.

Answer 3

Seriously? Bohr's Atomic Model? It was known by 1920 that that model failed (badly) to explain observational data. That is, it hasn't been taken seriously for nearly 100 years. Perhaps you should invoke the 4 humors and reason from there. (The only atom the the theory did a pretty good job for is hydrogen. One down, 91 elements to go....) There are lots of different models which are useful for different purposes. One thing we know about all of them, is they're all wrong. So, since the "right" model isn't known, picking a "best" model requires you to know what your purpose is, what the context is. The issue is deciding which model is most useful, not which model is correct. That turns out to depend on context. What I was taught as an undergraduate chemistry major (almost 50 years ago) was that an electron bound in an atom (or molecule, or ion, or radical) is best thought of as a probability wave. This means that the location of the electron is "smeared out" in space and that for any distance, r, from the nucleus (center), there is a non-zero probability of the electron being at that distance. This means, to answer your question, that atoms do not have "hard surfaces". The reason why some Physical Chemistry textbooks still cover the Bohr Model is because it helps the student transition between the physics of everyday objects to the non-intuitive world of quantum mechanics. The best book for you depends on what your mathematical knowledge is, how much effort you want to spend, and how detailed you want to be. It is a mature science, and many popular books have been written about it. Concepts such as "radius" are mostly nonsensical at the quantum scale. For instance, the radius of an atom depends on how you measure it. Consider the simple case of shooting electrons at it. Then shooting protons at it. Then shooting neutrons at it. Then microwaves. Then (visible) light waves. Then x-rays. Each will give you a different radius. In some ways it is similar to the question of how long is a coastline. If you measure it with a 100 meter cable, you'll get one answer (ignoring differences in sea level, tides, wind, rain, etc.), if you measure it with a meter stick, you'll get a larger value. Measure it centimeter by centimeter, and it will be even larger, and just imagine if you could measure it atom-by-atom, it would be very difficult to determine where the liquid began and where the land ended. Coastline is a macroscopic concept, with little meaning at the micrometer scale (let alone the nanometer scale). Very much like your concept of radius. (Although to be fair, we "steal" the word and use it in various contexts which often have similarities to its macroscopic use.)