In the following, take and for nitrogen gas:

Question

(a) Three moles of nitrogen at 30°C, contained in a rigid vessel, is heated to 250°C. How much heat is required if the vessel has a negligible heat capacity? If the vessel weighs 100 kg and has a heat capacity of how

much heat is required?

(b) Four moles of nitrogen at 200°C is contained in a piston/cylinder arrangement. How much heat must be extracted from this system, which is kept at constant pressure, to cool it to 40°C if the heat capacity of the piston and cylinder is neglected?

Answer 1

a. At constant volume, the total heat input will be (for constant heat capacity – if the heat capacity were not constant, we would have to do an integral). So, we have Q = 3 moles? 20.8 J/(mol K)? 220 K = 13728 J, to heat only the gas. To heat the vessel, with a total heat capacity of

100 kg? 500 J/(kg K) = 50000 J/K requires Q = 50000 J/K? 220 K = J. The energy required to heat up the vessel

is roughly 1000 times larger than the energy required to heat only the gas. This is fairly typical for gases: the vessel that contains them has greater mass, and therefore greater heat capacity, than the gas itself.

b. At constant pressure, (again for constant heat capacity; if the heat capacity were not constant, we would have to do an integral). So, we have Q = 4 moles? 29.1 J/(mol K)? –160 K = –18624 J. About 19 kJ has to be removed.

Note that in both parts of this problem, we only needed to know the total heat capacity (number of moles and molar heat capacity) and did not need to know the actual pressure or volume.