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Manicómio vs. manicômio vs. hospício

The Portuguese words manicômio, manicómio, and hospício all refer to an institution for the care of people with mental illnesses. The main differences lie in regional spelling rules (Brazil vs. Portugal) and historical or metaphorical usage.

Manicómio

B2
This is the Brazilian Portuguese spelling for a psychiatric hospital or asylum. The circumflex accent (^) indicates a closed vowel sound, which is standard in Brazil before nasal consonants (m or n). It is often used in historical contexts or when discussing the anti-asylum political movement.
O governo decidiu fechar o antigo manicômio da cidade.
(The government decided to close the city's old asylum.)
Muitos pacientes viveram naquele manicômio por décadas.
(Many patients lived in that asylum for decades.)
A luta contra o manicômio resultou em novas leis de saúde mental.
(The fight against the asylum resulted in new mental health laws.)

Manicómio

B2
This is the European (Portugal) and African Portuguese spelling for the same word. The acute accent (´) indicates an open vowel sound, which is the standard pronunciation in these regions. It carries the same meaning as the Brazilian version.
Este edifício serviu de manicómio no século passado.
(This building served as an asylum in the last century.)
Parece que estou num manicómio com tanto barulho nesta casa.
(It feels like I am in a madhouse with so much noise in this house.)
Os registos do manicómio foram perdidos no incêndio.
(The asylum records were lost in the fire.)

Hospício

B2
Originally meaning a place of shelter or hospice, this word became a specific term for an asylum. It is largely considered archaic or derogatory in medical contexts today. However, it is very commonly used in daily speech metaphorically to describe a chaotic, crazy, or disorganized situation (a madhouse).
Minha escola virou um hospício durante o intervalo.
(My school turned into a madhouse during the break.)
Antigamente, qualquer pessoa diferente era enviada para o hospício.
(In the past, anyone who was different was sent to the asylum.)
Trabalhar é um hospício, ninguém sabe o que fazer.
(Working there is a madhouse; nobody knows what to do.)

Summary

The difference between manicômio and manicómio is purely orthographic: the former is used in Brazil, and the latter in Portugal and Portuguese-speaking Africa. Hospício is an older term that applies to both regions; while it historically meant asylum, today it is predominantly used in informal speech to describe a chaotic place or situation.