English uses words from other languages and when they first took those words they kept the same pronunciation. You will find that most of the irregular words are from a different language.

The source of the /k/ sound of the digraph "ch" in "chem-", versus the /tch/ sound "ch" typically has in English (but cf. "character") is, as stated in previous answers, the etymology of the “chem-“ series: it comes from classical Arabic ("al") "kimiyá", itself derived from an earlier word for “Egypt” pronounced /k'em/ (NOT from Greek “heméia/huméia”, as often assumed!). Through Spain, (al) "kimiyá" passed into mediaeval Latin, Old French, and eventually reached English ("alchemist", "chemist", "chemistry", "chemic(al)", etc.), but too late to be affected by the rule that words written "ch" (originally "c" plus a front high vowel) are pronounced /tch/. So, we have /kémist/ instead of */tchémist/. As to "arch", observe that the digraph "ch" has two pronunciations, the native one /tch/ (cf. “arch”, “starch”, “church”, “torch”, [=syllable-final]…“chap”, “cheap”, “chop”…[syllable initial]), and the /k/ sound in the “chem-“ series and in ALL Greek words (“anchor”, "archaic", "archaeology", "archetype", "architect", "archive", “character”, “charisma” …."monarch(y)") that came into English in the 16th century or later. It only has the “native” /tch/ sound in "arch-bishop", "arch-deacon", "arch-duke", "arch-enemy", "arch-fiend", i.e., when it is a PREFIX AND it was adopted in the Old English period. In the case of “arch”, we find it written “arce-” in Old English texts of the 9th century, but surely existed since the seventh. Although, historically, “arch-“ comes from the same family of Greek words that gave us “archi[tect]” and “[mon]archy” (“arh(i)(éuo)” meaning “rule”, “reign”, as well as “old”, “origin”….etc.), the crucial fact was that whereas “archaic”, monarchy”…etc. came into English too late to be affected by the Old English rule “c”/k/+”e/i/0” -> /tch/, “arc(e)” was already English in time to be affected by the Old English palatalization. Thus, the different pronunciation of “ch” in the “chem-“ series and in “arch-” IS subject to RULE, after all.

Yes, it depends on the word's source, as well as how we came across it. The Latin word which gave us "arch" had a hard K sound (arcus), but came to us through French (arche), which had a "sh" sound. Then we changed it. We also gained the word "chemist" through French (yes, the French word also has a soft "sh" sound), but in that case we looked at its Greek origin - a word the Arab culture later borrowed - which had the "kh" fricative, which is often adapted to a K sound in English (viz. "chronology" and "archaic").

chemist is origin of arab word which pronounce like “k”。 i think arch shall have pronounced like “k” too origin of latin word， something happened in its evolution，someone incorrectly relay an erroneous pronunciation like “ch” ，“ch” is accepted through common practice from then on。

Eliot - so in this case there is no grammatical rule?