Nearly all Physics textbooks contain essentially the following statement (copied here from a text in the Microsoft Encarta Encyclopedia). Electrons that circle close to the (atom's) nucleus have less energy than electrons in orbitals farther from the nucleus.
The true situation is exactly OPPOSITE that claim!
The facts here are very simple and obvious. A basic law of science called the Conservation of Energy is what is involved.
For non-Physicists, it might seem most obvious if we first discuss a very similar gravitational situation such as a bunch of planets orbiting the Sun. Let's start with one planet which is orbiting in perfectly circular orbit at the distance of the Earth around the Sun.
Such a planet has Kinetic Energy due to the velocity of revolution orbiting the Sun, that is, for the Earth, Ek = 1/2 * mE * VE2, by the standard formula.
Or more generically
If we somehow moved that planet out to an infinite distance from the Sun, the planet would have ZERO Kinetic Energy of orbiting the Sun, because the velocity of revolution would drop to ZERO, per Newton's Gravitational formula
Going the other way, inward toward the Sun, if we somehow moved that planet in to distance of 1/10 as far from the Sun, the planet revolve at extremely high revolutional velocity and it would have enormous Kinetic Energy of orbiting the Sun, because the velocity of revolution would be very fast.
We can also analyze the Potential Energy of the Earth or any other planet or orbiting body. Per Newton, the only requirement for this is that the Gravitational Field be Conservative, which essentially means that the PATH between the two locations is unimportant, and only the LOCATIONS of the starting and ending points matters. The amount of WORK which needs to be done to move the orbiting body from one radius orbit to another is the definition of the amount of Potential Energy difference in the two situations.
We need to examine this with some simple Calculus.
Since the FORCE of the gravitational field changes with location, we must first look at the DIFFERENTIAL of the WORK, as it depends on the FORCE VECTOR.
We then need to MATHEMATICALLY INTEGRATE this DIFFERENTIAL over the range of locations, starting to ending, that is, from the STARTING orbital radius to the FINAL orbital radius.
We know that
F = G * M * m / d2
The Solution to the Integral is therefore:
(for the starting and ending locations).
Potential Energy is just the difference between these two:
We can now add both types of Energy, for two different locations. For an infinite distance, this means the one Kinetic term is zero because the velocity is zero, since the dimension d is infinite.
If we also define Potential Energy as being zero for an infinite distance, then we have:
Potential Energy for any real location is therefore always negative and kinetic energy is always positive.
In order to Conserve Energy, we must therefore describe a positive type of Kinetic energy and also a NEGATIVE type of energy called Potential Energy.
As we might bring such a planet closer to the Sun, the amounts of both types of energy change, but the total always stays constant, what is called Conservation of Energy.
As the planet might get very close to the Sun, like Mercury is, it would have LESS Potential Energy (because of the smaller orbital radius) and therefore it MUST also have MORE Kinetic Energy, because of the high orbital velocity.
As a result of this, Mercury has high orbital speed and a lot of Kinetic Energy, due to it's smaller orbital radius.
This shows the entire set of math formulas which show the Calculus reasons for why the amount of Work is always given by the standard formula, and which is then also given by the standard Potential Energy formula.
The point is that it is NOT distant Pluto that has a lot of Kinetic Energy, because of a bigger orbit, but instead Pluto has minimal Kinetic Energy for having very slow kinetic energy due to very slow orbital speed, which gives up most of it energy into Potential Energy.
Planets that circle close to the Sun or a star have MORE (Kinetic) Energy than planets in orbitals farther from the star.
Here are the formulas for the Earth which show that the average orbital or outward velocity is related to the average Potential Energy and therefore the average radius of the Earth's orbit. The final equation of this set also describes something called an Excape Velocity. IF these numbers are actually about the Earth and not the Sun, this calculates the exact speed needed by something on the surface of the Earth to escape to outer space.
For us on the surface of the Earth, these are the actual numbers which identify what our Escape Velocity is.
In other words, the electrons which circle CLOSEST to an atomic nucleus have the GREATEST amount of Kinetic Energy but they also have the least amount of Potential Energy!
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