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One of the free-living anaerobic nitrogen fixers is
(a)Rhodospirillum
(b)Rhizobium
(c)Azotobacter
(d)Beijerinckia

Answer
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Hint: They are gram-negative, motile, and spiral-shaped bacteria. The bacterium uses anaerobic fermentation or photosynthesis to produce energy as well as photoautotrophic growth.

Complete answer:
Nitrogen-fixing bacteria are those bacteria that fix the atmospheric nitrogen and make it available for the plants. Mainly, there are two types of nitrogen-fixing bacteria; they are free-living or non-symbiotic bacteria and non-free living or symbiotic bacteria. Examples of free-living bacteria include cyanobacteria, Azotobacter, and Clostridium. Examples of non-free living bacteria include Rhizobium and Bradyrhizobium.
Rhodospirillum is a free-living gram-negative, nitrogen fixer, and facultative anaerobe, so they can grow under many different types of conditions including aerobic or anaerobic environments. Rhizobium is a gram-negative, symbiotic, nitrogen-fixing soil bacteria that form an endosymbiotic relationship with roots of legumes. Beijerinckia is a free-living, aerobic nitrogen-fixing bacteria. Azotobacter is a gram-negative, free-living, aerobic nitrogen-fixing bacteria.

Additional Information: -Rhodospirillum bacteria can generally be found in marine environments, mud, and soil where light is available for photosynthesis.
-Another interesting character of Rhodospirillum was its purple color. It is a non-sulfur bacteria, which can evolve molecular hydrogen catalyzed by nitrogenase under nitrogen-deficient conditions using light energy and reduced compounds.
-Rhodospirillum can be used in alcoholic fermentation under low oxygen conditions or aerobic respiration in aerobic conditions.
-It is also used for the production of biological hydrogen fuel.
-It has the ability to live under cellular respiration.
So, the correct answer is ‘Rhodospirillum ’.

Note: -Free-living bacteria mean that they don't need to create symbiotic relationships with plants to survive and replicate. This happens in many plants like corn, they don't create symbiotic relationships with nitrogen-fixing bacteria.
-Important species include Rhodospirillum centenum, Rhodospirillum rubrum, Rhodospirillum photometricum.