how does light intensity affect the rate of photosynthesis?

Sometimes photosynthesis is limited by the concentration of carbon dioxide in the air. Even if there is plenty of light, a plant cannot photosynthesise if there is insufficient carbon dioxide.

how does temperature affect the rate of photosynthesis?

In practice, any one of these factors could limit the rate of photosynthesis.
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BBC - GCSE Bitesize: Factors limiting photosynthesis

There are three main factors which can limit the speed of photosynthesis
Light Intensity
Carbon Dioxide Concentration
A limiting factor is the component of the reaction (photosynthesis in this case) that is in shortest supply, and therefore sets a limit on the rate of reaction.
How light affects photosynthesis

Bright light is a basic element of photosynthesis
, but
in the color of light have an effect on plants.

Photosynthesis limiting factors ppt - TES Resources

The action spectrum of photosynthesis is a graph showing the rate of photosynthesis for each wavelength of light. The rate of photosynthesis will not be the same for every wavelength of light. The rate of photosynthesis is the least with green-yellow light (525 nm-625 nm). Red-orange light (625nm-700nm) shows a good rate of photosynthesis however the best rate of photosynthesis is seen with violet-blue light (400nm-525nm).

In practice, any one of these factors could limit the rate of photosynthesis.
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limiting factors photosynthesis Study Sets and …

If the light intensity is not a limiting factor, there will usually be a shortage of NADP+ as NADPH accumulates within the stroma (see light independent reaction). NADP+ is needed for the normal flow of electrons in the thylakoid membranes as it is the final electron acceptor. If NADP+ is not available then the normal flow of electrons is inhibited. However, there is an alternative pathway for ATP production in this case and it is called cyclic photophosphorylation. It begins with Photosystem I absorbing light and becoming photoactivated. The excited electrons from Photosystem I are then passed on to a chain of electron carriers between Photosystem I and II. These electrons travel along the chain of carriers back to Photosystem I and as they do so they cause the pumping of protons across the thylakoid membrane and therefore create a proton gradient. As explained previously, the protons move back across the thylakoid membrane through ATP synthase and as they do so, ATP is produced. Therefore, ATP can be produced even when there is a shortage of NADP+.

Photosynthesis and limiting factors

Plants cannot photosynthesize if it gets too hot.
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How temperature affects photosynthesis
The higher the temperature then typically the greater the rate of photosynthesis, because the activity of enzymes increases.

Factors Affecting The Rate Of Photosynthesis Biology ..

As we can see, there is a close relationship between the action spectrum and absorption spectrum of photosynthesis. There are many different types of photosynthetic pigments which will absorb light best at different wavelengths. However the most abundant photosynthetic pigment in plants is chlorophyll and therefore the rate of photosynthesis will be the greatest at wavelengths of light best absorbed by chlorophyll (400nm-525nm corresponding to violet-blue light). Very little light is absorbed by chlorophyll at wavelengths of light between 525nm and 625 (green-yellow light) so the rate of photosynthesis will be the least within this range. However, there are other pigments that are able to absorb green-yellow light such as carotene. Even though these are present in small amounts they allow a low rate of photosynthesis to occur at wavelengths of light that chlorophyll cannot absorb.