A Simple Mathematical Model First, Assumptions are an essential
part of all model development - Heater power is constant and is
inside the container so the gas begins heating up immediately. -
The gas mixes rapidly so that it heats uniformly - The gas is an
ideal diatomic gas (like air) so the molar heat capacity at
constant pressure is (Cp=7/2R) & (PV=nRT is the equation of
state) - The gas initially has a pressure, temperature, and
volume (Po,To, Vo) of 1.0 atmosphere (1.013x105 Pa), 300K, and
1.0 liters (0.001 m3). (initial conditions or boundary
conditions must alway be considered). Power = rate that Heat
Energy is added = nCp(dT/dt) This has the solution T = 300K +
Power *time/(nCp) using the equation of state, n=PoVo/(RTo), and
Cp=7/2R gives, T = To[1 + 2Power*time/(7PoVo)]. Finally using
the equation of state again V=nRT/Po Units are also very
important in mathematical models. In this example Power should
be in Watts, time in seconds, Pressure in Pascals(Pa),
temperature Kelvin, and Volume in cubic met