Why is some amount of air is left inside in your lungs after inhalation

Residual volume (RV) is the volume of air remaining in the lungs after maximum forceful expiration. In other words, it is the volume of air that cannot be expelled, thus causing the alveoli to remain open at all times. The residual volume remains unchanged regardless of the lung volume at which expiration was started. The residual volume (RV) is the amount of air that is left after expiratory reserve volume is exhaled. The lungs are never completely empty: There is always some air left in the lungs after a maximal exhalation.

So your answer will be Residual volume...

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It is left because exhaling will never remove all air from your lungs, if it did your body would shut down due to lack of oxygen to all organs. It distributed the atoms throughout the body.