Avogadro’s number is a crucial constant in chemistry and physics, representing the number of atoms in a theoretical specimen. However, determining the value of this number is not as simple as it may seem. In this article, we will explore the value of Avogadro’s number and why it should be a perfect cube.

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## An Integer Value for Clarity and Simplicity

Avogadro’s number should ideally be an integer, avoiding the need to interpret fractions of atoms. This is a reasonable expectation, especially for educational purposes. Fortunately, the currently accepted range for Avogadro’s number is (6.0221415 ± 0.0000010) × 10^23.

## The Physical Significance of a Perfect Cube

In envisioning the object containing Avogadro’s number of atoms, it is reasonable to imagine it as a perfect geometrical cube. This is because volumes of objects are typically measured in cubic units. With this in mind, the value chosen for Avogadro’s number should also be a perfect numerical cube.

## The Rarity of Perfect Cubes in the Estimated Range

Within the current range of acceptable integers for Avogadro’s number (approximately two hundred quadrillion or 2 × 10^17), only 10 of them are perfect cubes. These values range from 84,446,884^3 to 84,446,893^3. For our purposes, any one of these 10 values could be used. However, the one that is closest to the best current estimate of Avogadro’s number is NA* = 602,214,141,070,409,084,099,072, which is equal to 84,446,888^3. This value is accurate to within one unit in the eighth significant digit of the current best estimate.

## Establishing a Definitive Value

Our proposal is to permanently define Avogadro’s number, similar to the definitions of the speed of light and the second. We suggest setting Avogadro’s number equal to the specific integer NA*, as mentioned above. If the sides of the cube were even slightly longer or shorter, the number of atoms it contains would fall outside the currently accepted range.

## The Importance of Shape

The shape of the defining volume plays a significant role in determining the number of molecules it can contain. Considering this, it is natural to choose a cube as the defining object. While other solid shapes could also work, using a cube offers simplicity as it requires specifying only one side length. Choosing a rectangular solid or a sphere would introduce additional complexity or irrational numbers that are not ideal for this purpose.

## The Case Against 602,214,150,000,000,000,000,000

Although an alternative value that lies precisely in the middle of the current range could be considered, such as 602,214,150,000,000,000,000,000, it lacks physical significance. This value is neither a perfect cube nor a perfect square, making it impossible to construct a perfect geometrical cube or square of atoms with that exact volume or area.

## Ensuring Stability and Robustness

Choosing the most recent best estimate of Avogadro’s number as a fixed value is not a robust approach, unlike the methods used for defining the speed of light and the second. If experimental estimates of Avogadro’s number increase significantly, the “current best estimate” method would require adjusting the value accordingly. In contrast, the fixed values for the meter (defined in terms of the speed of light) and the second (defined in terms of cesium atom vibrations) are insensitive to minor refinements in the measurements.

## The Optimal Solution: An Integer Cube

In the same spirit as the fixed values for the meter and the second, the proposed value for Avogadro’s number, NA*, is a nearest-integer solution. It represents the nearest integral side length of a cube containing Avogadro’s number of atoms. This choice ensures stability and is largely immune to improved experimental estimates of NA. While the requirement of a perfect cube adds aesthetic appeal, it also holds practical and intuitive physical significance.

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