A Simple Mathematical Model

Here we use a simple example from introductory chemistry or physics as it should be familiar to many science educators. Assume that a gas is heated in a sealed flexible container so the pressure remains constant (P=P_o) and you want to develop a mathematical model describing how the volume of the container changes with time.

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 (C_p=7/2R) & (PV=nRT is the equation of state)
• The gas initially has a pressure, temperature, and volume (P_o,T_o, V_o) of 1.0 atmosphere (1.013x10⁵ Pa), 300K, and 1.0 liters (0.001 m³). (initial conditions or boundary conditions must alway be considered).

The controlling differential equation of this system is:

Power = rate that Heat Energy is added = nC_p(dT/dt)

or (dT/dt)=Power/(nC_p)

This has the solution T = 300K + Power *time/(nC_p)

using the equation of state, n=P_oV_o/(RT_o), and C_p=7/2R gives,

T = T_o[1 + 2Power*time/(7P_oV_o)].

Finally using the equation of state again V=nRT/P_o

V=V_o[1 +2Power*time/(7PoVo)]

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 meters. Although in the final result the first V_o can be in liters while the V_o inside the square brackets must be 0.001 m³.