A green plant synthesizes glucose by photosynthesis, as shown in the reaction $$6 \mathrm{CO}_{2}(g)+6 \mathrm{H}_{2} \mathrm{O}(l) \longrightarrow \mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}(s)+6 \mathrm{O}_{2}(g)$$ Animals use glucose as a source of energy: $$\mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}(s)+6 \mathrm{O}_{2}(g) \longrightarrow 6 \mathrm{CO}_{2}(g)+6 \mathrm{H}_{2} \mathrm{O}(l)$$ If we were to assume that both these processes occur to the same extent in a cyclic process, what thermodynamic property must have a nonzero value?

Photosynthesis is the process by which green plants, algae, and some bacteria convert sunlight, water, and carbon dioxide into glucose and oxygen. This conversion is represented by the chemical equation: \[6 \mathrm{CO}_{2}(g) + 6 \mathrm{H}_{2}O(l) \longrightarrow \mathrm{C}_{6}\mathrm{H}_{12}\mathrm{O}_{6}(s) + 6 \mathrm{O}_{2}(g)\].

This process is the cornerstone of life on Earth as it provides the organic compounds and oxygen necessary for the survival of most organisms. In the context of thermodynamics, photosynthesis is an endothermic reaction, meaning it absorbs energy, specifically from the sun, to proceed. While it appears to defy the tendency of systems to maximize entropy, it's driven by the constant influx of solar energy.

Glucose metabolism includes various biochemical processes that break down glucose to generate energy for cellular activities. Animals metabolize glucose in a series of reactions known as cellular respiration, particularly glycolysis, the citric acid cycle, and oxidative phosphorylation. The equation \[\mathrm{C}_{6}\mathrm{H}_{12}\mathrm{O}_{6}(s) + 6 \mathrm{O}_{2}(g) \longrightarrow 6 \mathrm{CO}_{2}(g) + 6 \mathrm{H}_{2}O(l)\]
represents the catabolism of glucose where it combines with oxygen, leading to carbon dioxide, water, and the release of energy in the form of ATP. This exothermic reaction is essential for maintaining the life processes of organisms that are not capable of photosynthesis.

Thermodynamics is the study of heat and energy transfer, conversion, and the work done by or on systems during these processes. It's summed up by its laws, with the first law indicating the conservation of energy and the second law stating that the entropy of an isolated system always increases over time. In the context of our reactions, the first law relates to the conservation of energy between photosynthesis and glucose metabolism, while the second law guides the spontaneous flow of the reactions based on the increase of total entropy, albeit locally decreased during photosynthesis.

Entropy measures the disorder or randomness of a system and is a fundamental aspect of the second law of thermodynamics. An increase in entropy reflects an increase in disorder and energy distribution. While both photosynthesis and glucose metabolism influence the system's entropy, the overall entropy change of the universe will always be positive, according to the second law. This principle allows us to understand why there is a continuous requirement for energy input, such as from sunlight, to maintain the cyclic order of processes like photosynthesis.