Calorimeter Heat (q = C_cal ΔT)

Q = mcΔT works when you can weigh the thing being heated and look up its specific heat. A bomb calorimeter is a steel pressure vessel, a water bath, a stirrer, a thermometer and an ignition wire, and nobody weighs that assembly or looks anything up. Instead the whole apparatus is calibrated as a single number — the calorimeter constant C_cal, in joules per kelvin — and then heat is just C_cal times the temperature rise.

The calibration is done by burning something whose heat of combustion is certified: benzoic acid, at 26.434 kJ/g, is the international standard for the job. Burn 1.000 g of it and watch the temperature climb 2.475 C°, and C_cal = 26 434/2.475 = 10 680 J/K. That number belongs to that instrument, with that much water in the bucket, and it has to be redetermined if anything about the assembly changes.

The sign trap is worth stating plainly: this q is the heat that went into the calorimeter, not the heat of the reaction. The reaction's own heat is the negative of it. A rising temperature means the calorimeter gained energy, which means the reaction lost it, which means the reaction is exothermic and its q is negative. Reporting this figure with the sign left on it is the standard way a combustion enthalpy comes out backwards and positive.

One more subtlety separates a bomb from an open cup. A bomb holds volume constant, so what it measures is ΔU, the internal energy change. An open coffee-cup calorimeter holds pressure constant and measures ΔH. The two differ by the work of pushing the atmosphere aside, ΔH = ΔU + Δn_gas·RT, which for a reaction changing its gas moles at room temperature is a couple of kJ per mole — small, but larger than the precision of a good bomb.

Dividing this heat by the moles burned turns it into a molar enthalpy, which is the number that goes in a table. Doing that division with the mass instead of the moles gives a specific energy in J/g, which is what a fuel is sold on.