There are heterotrophic prokaryotes, and there are autotrophic prokaryotes. In the autotrophic prokaryotes category, there are photoautotrophic prokaryotes and chemotrophic prokaryotes.
Are there chemotrophic eukaryotes?
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The First classification is very much based on a complete segregation between respiration and photosynthesis. In the first classification, anything that needs another animal to get food is a 'Heterotroph'. (Here, chemotroph and chemoautotroph are the same things)
But Generally the 2nd Classification is preferred, because prokaryotes have some very unique nutritional habits that cannot be explained on the basis of 'Eating/Not Eating'
In a way synthesis is the creation of a carbon compound by use of some energy. So we need to classify based on energy and carbon source. Here organic carbon source means it is a 'hetero'troph and inorganic means it is an 'auto'troph. Similarly if Light energy is the direct energy required, it is a 'Photo'troph and if a chemical compound is required it is a 'Chemo'troph
By the First Classification, there are no 'Chemotrophic/Chemoautotrophic eukaryotes' meaning no eukaryote can produce food from inorganic material using chemical energy.
But by the 2nd Classification, essentially all eukaryotes that do not photosynthesize, instead use organic compounds as a food source, all animals, fungi etc. are 'Chemoheterotrophs'*However no Eukaryote is a 'Chemoautotroph'