Statement I is correct: Energy flow in an ecosystem is unidirectional. It originates from the sun, is captured by producers (photosynthesis), and then flows through different trophic levels (consumers). Energy is lost as heat at each level, so it cannot cycle back to producers in the same way nutrients do.
Statement II is incorrect: Ecosystems are \textit{not} exempt from the laws of thermodynamics. The second law, in particular, is relevant: energy transformations are never 100\% efficient, and some energy is always lost as heat. This explains the unidirectional flow of energy in ecosystems.