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In the textbook,it is said that bomb calorimeter measures the constant volume heat ,while the cup calorimeter measure Constant pressure Heat.I was wondering, why cup calorimeter is considered constant pressure,even though it also does not allow a gas to expand (it has a fixed volume).So if not then that means cup calorimeter allows some air to get in so that pressure is constant ?(I.e the system is not closed )?

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Even though the pressure remains constant, heat can get in (or out) of the system. From the cup calorimeter, we are trying to find $\Delta H$ which is equal to the heat evolved from the system at constant pressure. We measure this heat and our job is done.

$$\Delta H=q_p$$

If we want to calculate $\Delta U$, we just to find the heat evolved from the bomb calorimeter, which can be done by measuring the change in calorimeter's final and initial temperature given that the heat capacity of the calorimeter is known.

$$\Delta U=q_v=nS\Delta T$$

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