4.5. As you are no doubt aware, we in the academic world are most anxious to prepare you to solve technical problems that you may encounter in your future endeavors. Our Advanced Planning Section has been examining some unusual projected problems and requests your assistance on the particular one described below. Decades from now, the present method of supplying energy to households (i.e., with electricity, gas, or oil) may not be possible. Instead, housepersons will shop for their energy in supermarkets (Figure P4.5). Cylinders of gas (let us assume that the cylinders contain air) may be purchased and connected to any number of Carnot engines or other such efficient devices to be stocked in the home. Work is then obtained which may be utilized by the houseperson's family. When purchased, the cylinders are packed in well-insulated bags which may be removed (if desired) when connecting to a work-producing device. The problem we face is to devise a convenient method to allow the houseperson to compare prices for the various gas cylinders available at the supermarket. The usual size for most cylinders is 1 m, but the initial air pressure and temperature vary widely. One produced by R. Jones, Jr. is widely advertised to be quite economical, but our analysis indicated that the cylinder contained no air at all! R. Nader III is expected to object to this deplorable situation and to require that we provide a simple equation to allow housepersons to calculate quickly (on their HP-1001 or equivalent) the unit cost of work energy in joules per dollar knowing only the initial temperature (K) and pressure (N/m) of the air in the cylinder as well as the selling price of the cylinder. Data: Assume that the ambient temperature and pressure are 300 K and 1 x 105 N/m. Air has a heat capacity at constant volume of 20.7 J/mol K, and, at constant pressure, of 29.0 J/mol K. The gas constant is 8.314 J/mol K and the gravitational acceleration is 9.81 m/s. Derive an equation for the unit cost of produced work and demonstrate its application for a cylinder 1 m in volume initially at 8 x 105 N/m and 400 K which sells for $0.32.