Determine the ASTM grain-size number if 25 grains per square inch are measured at a magnification of \(75 \times\).

Grain size measurement is a crucial process in material science, particularly in metallurgy, where it affects the physical properties of a material like strength, toughness, and ductility. In the microscopic examination of a polished and etched sample, grains are the distinct crystals that make up the metal. To measure grain size, scientists count the number of grains within a specific area.

However, because different magnifications can make it challenging to maintain consistency, a standard magnification (100x) is used to ensure comparability of results. At this standard magnification, one can compare the grain sizes by counting the number of grains within a square inch. This data can then be utilized for various calculations, including ASTM grain-size number, which gives a single number to represent the average size of the grains within a metal sample.

When working with microscopic images, magnification adjustment is key to accurately sizing and comparing features such as grains in a metal sample. Microscopes use lenses to magnify objects, and the magnifying power is expressed as a multiplication factor, like 75x or 100x, indicating that the image is 75 or 100 times bigger than the actual size.

Adjusting the magnification is not just about zooming in or out; it also directly impacts grain size measurements. The given exercise highlights the need to adjust grain counts to a standard 100x magnification from any other magnification level by using a specific mathematical relationship. This standardization is critical because comparing grain sizes at different magnifications without adjustment would lead to incorrect conclusions about the material's characteristics.

The American Society for Testing and Materials (ASTM) has developed a formula to standardize the calculation of grain sizes. This formula allows material scientists and engineers to express grain size as a number, which is the ASTM grain-size number (G). The ASTM grain-size formula is expressed as \(N = 2^{(G-1)}\), where \(N\) represents the number of grains per square inch at a magnification of 100x.

To determine the ASTM grain-size number, one must first adjust the grain count to the standard magnification, as previously mentioned, and then solve the equation for \(G\). In this formula, the number 2 represents the doubling nature of grain division. As grains grow larger, each increase in the ASTM number represents a doubling of the number of grains, hence indicating finer grains. Understanding and applying the ASTM grain-size formula is essential for characterizing materials and ensuring that they meet the necessary specifications for their intended application.