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A calorimeter measures enthalpy indirectly: the reaction warms a known mass of water, and Q = mcdT with c = 4.184 J/g/K recovers the heat. Dividing that heat by the moles reacted and flipping the sign gives the molar enthalpy change, since heat gained by the water was lost by the reaction.
Heat absorbed by water
Q = m x 4.184 x dT
Molar enthalpy
dH = -Q / moles reacted
Warming water means the reaction gave heat away, and the sign convention makes energy leaving the system negative.
Yes, and a careful experiment adds its heat capacity. This simple treatment attributes all the heat to the water, which slightly understates the enthalpy.