Why does "appropriate" (and also "duplicate", "deliberate" etc) have a different vowel in their adjective/noun and verb forms?

I have no reference for this, but the evidence seems reasonably convincing. It might be supposed that, as "-ate" is one of the rare suffixes that is not stress neutral (-IC, -ICS, -ATE, -AL, -AN, -OUS, -IVE), and therefore, as it imposes primary stress regularly on the antepenultimate of three or more syllable words, without distinction of the nature of the word, it was decided that the distinction that can be otherwise introduced by means of early stress and late stress, could be managed through vowel reduction; this appears even more so to be the case from the fact that there is no uniform reduction of the suffix "-ate"for two syllable -ate ending words (this reduction occurring not systematically but rather exceptionally for familiar two syllable -ate ending words (2/ and 3/ below)).

(LPD, p. 51, -ate) This suffix is regularly strong, eɪt, in verbs, but often weakened to ət ɪt in nouns and adjectives. Its influence on stress depends on the length of the word.
(1) In two syllable verbs stress usually falls on the suffix in BrE (vi'brate, cre'ate), but on the stem in AmE ('vibrate, 'create).
(2) In longer verbs, the stress generally falls on the antepenultimate ('demonstrate, dis'criminate, as'sociate). There are a few exceptions and cases where speakers disagree ('sequestrate or se'questrate).
(3) In nouns and adjectives the suffix is unstressed ('private, 'climate), and in longer words the primary stress generally falls two syllables back from the suffix ('delegate, 'vertebrate, ap'propriate; important exceptions are in'nate, or'nate, se'date). The suffix vowel is generally weak in familiar words ('climate, 'private), though in some words speakers vary ('candidate, 'magistrate). In more technical words a strong vowel is retained ('sulphate, 'causate).
(4) Note the distinction between verb and noun/adj in cases such as 'separate, as'sociate, 'moderate, 'delegate.

Most of the two syllable -ate ending words ("ət" between parentheses shows an exception to the more usual eɪt pronunciation)

• 1/ late stress, eɪt, only v.: ablate, collate, cremate, deflate, equate, fixate, formate, gestate, gradate, ligate, locate, migrate, narrate, negate, notate, nutate, orate, palpate, pulsate, pupate, relate, stagnate, titrate, vacate, vibrate,
  (not a suffix but only an ending: ablate, berate, cu'rate (eɪt, v.), dilate, instate)

• 2/ early stress, eɪt, no v.: chordate, curate (ət for the n.), magnate, prelate (ət), primate (ət (archbishop), eɪt (higher animal)), private (ət), senate (ət),
  (not a suffix but only an ending: cognate (ət), frigate (ət), ingrate, oblate, substrate, template)

• 3/ early and late stress according to n./a. and v. difference, eɪt: castrate, dictate, frustrate, gyrate (ət for the a.), hydrate, lactate, legate (ət for the n.)
  (not a suffix but only an ending: conflate, donate)

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Addition made by user Decapitated Soul in the way of providing more information about the suffix "-ate"

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Here's what English Suffixes - Stress-Assignment Properties, Productivity, Selection and Combinatorial Processes by Ives Trivian says:

This noun suffix (from L[atin] -ātus) rivals -ship in denoting offices or functions and -dom or -hood in naming institutions or collective bodies: caliphate, cardinalate, catechumenate (< catechumen), collectorate, consulate, directorate, doctorate, duumvirate (< L <~ duumvir + -ate), electorate, exarchate (= exarchy), governorate, imamate (< imam), in- spectorate, etc. (40 items). In the senses defined above, the -ate noun suffix has been far less productive than its Germanic rivals as it does not have the capacity to attach to native bases. It is probably no longer alive as it has not formed any noun since the end of the 19th century.

-ate is occasionally interchangeable with its rival form -ship in the sense of “position of ” (collectorate/collectorship, pastorate/ pastorship, professorate/professsorship, rectorate/rectorship). In the foregoing noun class the -ate suffix may additionally denote members of a function collectively, a sense which -ship does not have: pastor- ate (s. above + “pastors collectively”); professorate (id. + “professors collectively”). There is no synonymy in the pairs below:

(5) _{consulship (“post of a consul”) ≠ consulate (“residence or workplace of a consul”); directorship (“post of a director”) ≠ directorate (“group of people directing a corporation”); electorship (“status of an elector”) ≠ electorate (“body of electors in a given district”); governorship (“office of a governor”) ≠ governo- rate (id. + “an administrative division of a country”); inspectorship (“office of an inspector”) ≠ inspectorate (“a body of inspectors”); protectorship (“position of a protector”) ≠ protectorate (“state or area controlled by another state”)}

In the meanings defined above, the separable -ate noun suffix is stress-neutral: 'cardinalate <~ 'cardinal, 'governorate < 'governor, 'mandarinate <~ 'mandarin, 'matriarchate < 'matriarch, 'patriarchate <~ 'patriarch, 'presbyterate <~ 'presbyter (cp. synchronically transparent or opaque verbs in -ate and their possible homographic nouns or adjectives, eg. separate, v. and adj., duplicate, v. and n.). In terms of stress-assignment, opaque-stem nouns in -ate denoting a function, institution, etc., will have to be treated together with -ate verbs: legate (+ v.), delegate (id.), etc. (§13.3.1).

-ate is also a noun suffix used in chemistry to designate an element of a compound, especially a salt or ester of an acid named by a noun syngtam with an adjective in -ic (in the same way -ite nouns are correlative to acids with an adjective in -ous, cf. §10.4): acetate <~ acetic acid, borate <~ boric acid, carbonate <~ carbonic acid, chlorate <~ chloric acid, hydrate <~ hydric acid, nitrate <~ nitric acid, sulphate <~ sulphuric acid, etc.

Whereas the foregoing nouns may be analysed as resulting from affix-substitution between -ic and
-ate, the -ate chemical suffix is always stress-neutral, as opposed to -ic (car'bonic/'carbonate), although it does not reduce: ['naɪtr eɪt], etc.

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End of addition due to user Decapitated Soul

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Note that the stress neutral property of "-ate", as related in the preceding addition, does not take into account the length of the words. In three syllable words stressed on the the second syllable or two syllable words stressed on the first the shift is already effected, so to speak; in many other cases there is a stress shift.

• 'pastorate, di'rectorate, pro'fessorate, e'lectorate, in'spectorate, pro'tectorate, 'doctorate, 'activate, af'fectionate, 'alienate, a'malgamate, etc.
• (however) an'tique/'antiquate, a'spire/'aspirate, 'calumny/ca'lumniate, 'certify/cer'tificate, 'college/col'legiate, con'dense/'condensate, de'spair/'desperate, 'difference/diffe'rentiate, di'stil/'distillate, e'xude/'exudate, 'habit/ha'bituate, 'humble/hu'miliate, i'dea/'ideate, etc.

For chemical terms this could be true, but a serious check is needed.

• 'hydrogen/'hydrogenate, 'oxygen/'oxygenate,