Heat capacity and calorimetry

How much does a substance warm up when heated? The heat needed is proportional to the rise in temperature: q = C∆T, where C is the heat capacity.

• Molar heat capacity, C_m = C/n: the heat needed to raise one mole by 1 K. So q = n × C_m × ∆T.
• Specific heat, c: the heat needed to raise one gram by 1 K. So q = m × c × ∆T.

For example, warming 100 g of water (c = 4.18 J g⁻¹ K⁻¹) by 10 K needs q = 100 × 4.18 × 10 = 4180 J.

Heat capacity can be measured at constant volume (C_V) or at constant pressure (C_p). At constant pressure, some of the heat also does work as the gas expands, so C_p is larger. For an ideal gas, the difference is exactly C_p – C_V = R.

Energy changes in reactions are measured in a calorimeter:

• A bomb calorimeter is a strong, sealed steel vessel (the 'bomb') in a bath of water. Since the volume cannot change, no expansion work is done, and the heat measured gives ∆U. The heat given out by the reaction is found from the temperature rise of the calorimeter: q = –C_V × ∆T.
• A coffee-cup calorimeter is open to the air, working at constant pressure, so it measures ∆H.

When a reaction neither makes nor uses up gas molecules (∆n_g = 0), as in C(graphite) + O₂(g) → CO₂(g), then ∆H = ∆U, and the bomb calorimeter result gives ∆H directly.