We lean the mixture as we climb because of the decrease in air density. We are trying to keep the ratio of fuel to air by mass constant, not by volume. The volume remains the same. The combustion chamber does not change in size. However, the decreasing air density, because of decreasing pressure, continues to decrease the number of air molecules that enter the combustion chamber. Say our proper fuel/air ratio is 1:1 at sea level. If we climbed to 10,000 feet and did not lean the mixture we might end up with a fuel/air ratio of 2:1.
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