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Respiration is basically a process of break down of complex carbohydrates while in this process energy is released which is trapped into ATPs, photosynthesis is quite the opposite of it, it is the process by which those complex carbohydrates are synthesized from simple compounds like water and carbon di oxide.
Cellular respiration, by definition is the process by which a cell breaks down glucose molecule into carbon dioxide and water. There are several very delicately interdependent biochemical cycles that are involved, lets look at them in brief.
The moment glucose molecule enters the cell, the first thing that is done is to TRAP it inside, by phosphorylation. This phosphorylated glucose molecule then enters glycolysis in the cytoplasm itself, the aim of glycolysis when oxygen is available is to synthesize pyruvate which will then enter the second stage of cellular respiration that is kreb's cycle. In the process the net gain of 2 ATPs occurs. Those cells which cant conduct kreb's cycle, thrive on these 2 ATPs for their energy need. (RBCs is an example)
The pyruvate thus synthesized now has to enter mitochondria for the kreb's cycle. The pyruvate needs to be converted to acetyl coA for this purpose, acetyl coA is one wonderful molecule which is a common gateway for many of energy yielding processes, oxidation of fatty acids is one example of those. There are many reactions that occur in kreb's cycle but out of these those reactions which generate reducing equivalents like NADH and FADH2 are most important. Kreb's cycle also liberates 2 carbon dioxide molecules.
The reducing equivalents mentioned above will now enter a complex process of oxidative phosphorylation in mitochondrial intermembranous space (the space between the two membranes of mitochondria) here these molecules (NADH and FADH2) are oxidised again but this process is closely linked with phosphorylation of ADP to ATP. Water is formed as the product.
This is how cells generate ATP from the breakdown of glucose molecule. This is cellular respiration.
I think this link will help.
Photosynthesis on the other hand is the reaction by which the complex carbohydrates are synthesized. We can equal this to a rechargeable battery which stores electrical energy for usage later on.
To simplify the matter, the light energy is trapped into the carbohydrate molecule, by the reaction which is overall a reduction reaction. This molecule (e.g Glucose) can be oxidised later to release this trapped energy.
Photosynthesis is a two stage process, the first stage involves formation of reducing equivalents called NADPH using light energy and these reducing equivalents later on used to "reduce" (it is not a proper expression for the reaction as this reaction is not strictly a reduction reaction but the concept is more clearly understood in this way) carbon dioxide into carbohydrates, using molecules of water in the process.
The first reaction i mentioned is light dependent, which can only occur in the presence of special pigments e.g chlorophyll, which absorb specific wavelengths of light. It is because these pigments that leaves (which are organs of photosynthesis and respiration in plants) appear green to us.
the overall simplifiled version of photosynthesis reaction is something like this
CO2 + water ----> carbohydrate molecule (e.g glucose) + oxygen
It is very interesting to note that oxygen is FORMED de novo in this reaction. It is postulated that all of the oxygen that is present in the atmosphere today is the product of photosynthesis of very very very ancient plants. We certainly owe a lot to our green friends out there.