en_tn/2ch/36/22.md

in the first year

This refers to the beginning of the reign of King Cyrus. (See: rc://en/ta/man/translate/translate-ordinal)

so that the word of Yahweh by the mouth of Jeremiah might be carried out

Here the noun "word" can be translated with the verb "spoke." The word "mouth" represents Jeremiah. This can be stated in active form. Alternate translation: "so that what Yahweh spoke through Jeremiah might happen" or "so that the word from Yahweh that Jeremiah spoke might happen" (See: rc://en/ta/man/translate/figs-synecdoche and rc://en/ta/man/translate/figs-activepassive)

Yahweh motivated the spirit of Cyrus, king of Persia

Motivating the spirit is a synecdoche for making someone want to act. Alternate translation: "Yahweh made Cyrus, king of Persia, want to act" (See: rc://en/ta/man/translate/figs-synecdoche)