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Pressure pumps are used to apply pressure to a system. What would happen if someone attempted to fill, for example, a bicycle tire (with a check valve) to 2 ATM of pressure using a pressure pump that could only supply 1.5 ATM of pressure? As the gas cannot go backwards due to the check valve during filling, could the pressure be expected to rise in the tire to 2 ATM?

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No, because the pump is designed to limits and if you exceed its working or maximum pressure then, for example, its seals fail and it just does not pump any more.

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