Homogeneous is an adjective that has several meanings.
Homogeneity has a precise meaning in physics.
In biology homogeneous has a meaning similar to its meaning in mathematics. Generally it means "the same" or "of the same quality or general property", such as a homogeneous sample, homogeneous population, etc.
Homogenous (without the second e) has a similar meaning of being the same throughout, and is perhaps more common in everyday speech. For example homogenised milk is milk which has been processed so that the cream is mixed throughout the milk, rather than being left to settle at the top.
In physical chemistry, homogeneous describes a single-phase system as opposed to heterogeneous where more than one thermodynamically distinct phases co-exist. Example: a sugar cube in a glass of water is bi-phasic and heterogeneous, sugar dissolved in water is a homogeneous system. See also phase diagrams and the classification of catalysts.
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