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Chapter 4: Problem 48

Determine the ASTM grain-size number if 30 grains per square inch are measured at a magnification of \(250 \times\).

Short Answer

Expert verified
Answer: The ASTM grain-size number for the given sample is approximately 0.

Step by step solution

01

Identify the given values

We are given the number of grains per square inch (N) as 30 grains, and the magnification (M) as \(250 \times\).
02

Calculate the number of grains per square inch at 100x magnification

According to ASTM standards, the grain-size number is calculated at 100x magnification. Therefore, we need to adjust the given number of grains per square inch to represent 100x magnification. We can do this using the following formula: $$ N_{100} = N(M_{actual} / M_{standard})^2 $$ where \(N_{100}\) is the number of grains per square inch at 100x magnification, \(N\) is the given number of grains per square inch, \(M_{actual}\) is the actual magnification, and \(M_{standard}\) is the standard magnification (which is 100x in our case). Plugging in our values: $$ N_{100} = 30(250 / 100)^2 $$
03

Calculate \(N_{100}\)

Now, let's calculate \(N_{100}\): $$ N_{100} = 30(2.5)^2 $$ $$ N_{100} = 30(6.25) $$ $$ N_{100} = 187.5 $$
04

Calculate the ASTM grain-size number

We can now calculate the ASTM grain-size number (G) using the following formula: $$ G = -6.64 + 2\log_{10}(N_{100}) $$ Plug in the calculated value of \(N_{100}\) from step 3: $$ G = -6.64 + 2\log_{10}(187.5) $$
05

Compute the ASTM grain-size number

Finally, let's compute G: $$ G = -6.64 + 2(2.27) $$ $$ G = -6.64 + 4.54 $$ $$ G = -2.1 $$
06

Interpret the result

The ASTM grain-size number we obtained is negative, which means that our sample has larger grains than the base size (G=0) according to ASTM standards. In practice, we would round our result up (to G=0), because a negative ASTM grain-size number isn't practically meaningful. So, the ASTM grain-size number for the given problem is approximately 0.

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