Particle to use with 順番

Question

I am writing a report in Japanese, describing how I conducted an experiment. The first two lines are (some information changed for NDA stuff):

Aを30ｇぐらい秤量し、すり鉢で細粒化しました。受皿、粉砕用ふるいの順番でふるい振とう器に乗せ、粉砕ボールと細粒化したAを入れ、蓋をしました。

I am pretty sure this conveys the meaning of:

[I] weighed ~30g of A, and grinded [it] in using a mortar and pestle. In the order of receiving tray, then screen for pulverizing, [I] placed [them] on the shaking machine. Then, [I] put in the pulverizing balls and the grinded OCP, and shut the lid.

Some of these more scientific terminology are hard translate succinctly in English, so I tried my best in that sense.

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Main Question:

When I looked up 順番 on Jisho, most of the sentences had 順番 using the particle に:

彼らは順番に歌を歌った。

They each sang in turn.

However, the use of で　comes from a procedure document my (native) coworker wrote. Both seem to me like they could work, に describing a general style of action while で　describing a more specific method. But I am not sure I am thinking about this correctly. What could be the reason for picking で　over に　in this situation?

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Second:

If I am correct in assuming that the 「受皿、粉砕用ふるいの順番で」phrase is describing a manner or method, is there an implied [them], like I have in my English translation, that is dropped due to context? Or is it not necessary since the "order" references the actual nouns, as opposed to something like "in increasing order".

Answer 1

順番に describes something being done in turns regardless, whilst 順番で describes something being done in turns and the order is important.

Regarding the given example under main question:

彼らは順番に歌を歌った。
They each sang in turn.

Each of them took a turn to sing. That tells everyone of them sang, regardless of whom was first, next, or last. In this example, the order is not known or does not matter.

However, the use of で　comes from a procedure document my (native) coworker wrote. Both seem to me like they could work [...] What could be the reason for picking で　over に　in this situation?

I would agree with the coworker that 順番で is indeed accurate. If the particle is substituted as it is and becomes 順番に, that seems not accurate. The preceding particle may be amended for a better sense, but the modified sentence would remain ambiguous.

受皿、粉砕用ふるいの順番で ○ (accurate)
受皿、粉砕用ふるいの順番に △ (not accurate)
受皿、粉砕用ふるいを順番に △ (ambiguous)

The last modified sentence tells that the plate and the screen, are set in order one by one, but the order is not known or assumed to have been told earlier (not true, because the order is not known unless using the particle で).

Regarding the second part:

If I am correct in assuming that the 「受皿、粉砕用ふるいの順番で」phrase is describing a manner or method, is there an implied [them], like I have in my English translation, that is dropped due to context? Or is it not necessary since the "order" references the actual nouns, as opposed to something like "in increasing order".

順番で would emphasize that the order is important. Therefore, the more understood translation would be:

受皿、粉砕用ふるいの順番で ふるい振とう器に乗せ [...]
[Set] the receiving plate, pulverizing sieve, in [this aforementioned] order, then place [them] onto the sieve shaker [...]

That means in this order: The receiving plate comes first (at bottom), then followed by the sieve (on top of plate), and so on.