Eli5 – How is the kilogram defined using the second and the Plank constant?


The kilogram used to be defined by a specific chunk of platinum-iridium.

But now it’s defined using the second and the Plank constant, somehow.

Assuming you already have a very accurate measurement of the second, and the Plank constant, how do you combine those two things to get the kilogram?

In: 1

Correction. Three constants combine to define the kilogram.

The second

The Plank constant

The meter

I still don’t understand how they combine.

Planck’s constant has units. It’s not just 6.626 x 10^(-34). It’s that many Joule Seconds.

A joule is a measurement/unit of energy, and it can be broken down into kilograms * meters^2 / seconds^2. When I say “joule seconds”, the exponent on the seconds under the fraction is reduced from 2 to 1.

Regardless, “kg” is now a unit you can solve for in a physics experiment using the above figures. Albert Einstein won the nobel prize for the photoelectric effect, which is essentially what solar panels run on and one of the easiest ways to use planck’s constant in action. So go calculate it.

So the Planck constant has, as part of it’s formula, a mass component because it involves the relationship between mass and energy.

Obviously, it’s a constant so it doesn’t change, but the unit that we use for the mass part can change what the number looks like — any calculation involving mass will have very different numbers if using a gram, an ounce, or a ton.

So what they did was express the definition of the Planck constant to be in kilograms. The other key parts of the equation are the second and the meter — and both of those are defined using universal constants as well.

The end result of this is that since neither the values for the Planck constant, the second, and the meter are consistent and can never change, that means the kilogram is too, because if you know the other three values, you can calculate what the mass of a kilogram is.

The Planck constant gives the relation between the frequency and the energy of a photon. One way energy can manifest itself is in accelerating heavy objects; one Joule is the energy needed to accelerate a mass of one kilogram by one m/s². The Planck constant being 6.62607015×10^(−34) Joules per Hertz thus means that a single photon of a frequency of one Hertz fired at a mass of one kilogram will accelerate that mass by 6.62607015×10^(−34) m/s² (note: this is a thought experiment, not a viable experimental setup).

So, the kilogram being defined through the Planck constant means that one kilogram is however much mass can be accelerated by 6.62607015×10^(−34) m/s² by one single photon of a frequency of one Hertz.

The meter, by the way, is defined in a similar way through the speed of light; one meter is defined as “however far light travels in 1/299792458 second in a vacuum”.

You can put a mass on a balance, and it will tip to that side. There are a couple of ways you could pull it back level. One way is to put an equal mass on the other side, so the force of gravity is equal on both sides. Or you could put some other kind of force on the other side. What if we used an electromagnet to pull it back to level? That’s what the watt (Kibble) balance essentially is.

So they carefully control the number of windings, thickness of wire, length of coil… Whatever needs to be controlled so that the only variables are the voltage and current running through the coil. Current is measured in amps which is equivalent to coulombs per second (we know the charge on an electron, so if we count the number of electrons passing per second, we’re measuring electric current). And volts is equivalent to joules per coulomb (again if we put an electron in a voltage field, the electron will want to move from negative to positive, that energy given to each electron to move is the measure of how strong the field is). If you multiply volts and amps together, you get watts. Notice that coulombs cancels out and you’re left with joules per second, and watts are equivalent to joules per second.

Now, a joule is the amount of energy it takes to lift a 1 newton object by 1 meter. So Joules are equivalent to newtons*meter. And with one more step we can break newton’s down into the core, SI units. A newton is a measure of force. From Isaac Newton, we learned that it’s the force required to accelerate a 1kg mass by 1m/s². And so a newton is equivalent to kg*m/s².

Let’s throw it all together. Watts are equivalent to J/s. Joules are equivalent to N*m. And newtons are equivalent to kg*m/s². Replace everything to get to the base, SI units and we have 1 kg*m²/s³ = 1W of power. Crazy right? If we can base meters and seconds off fundamental physics constants, we can turn around and define the kg based on the same units. Planck’s constant not being a unit is more or less there to make the math look nicer as I understand. The same way we use Avogadro’s number so we don’t have to talk about sextillions of things. (Note, Planck’s constant does *have* units, but it itself is not a unit)