One of the oldest units for atmospheric pressure is \(\mathrm{lb}\) in. \(^{-2}\) (pounds per square inch, or psi). Calculate the numerical value of the standard atmosphere in these units to three significant figures. Calculate the mass in pounds of a uniform column of water \(33.9 \mathrm{ft}\) high having an area of \(1.00 \mathrm{in} .{ }^{2}\) at its base. (Use the following data: density of mercury \(=13.6 \mathrm{~g} \mathrm{~mL}^{-1} ;\) density of water \(=1.00 \mathrm{~g} \mathrm{~mL}^{-1}\); \(1 \mathrm{~mL}=1 \mathrm{~cm}^{3} ; 1 \mathrm{lb}=454 \mathrm{~g}\) $$ \text { in. }=2.54 \mathrm{~cm} .) $$

Understanding the concept of a standard atmosphere is foundational when dealing with atmospheric pressure calculations. The standard atmosphere is a unit of pressure defined as 101325 pascals (Pa). This value is also roughly equivalent to one bar, another common unit of pressure. To visualize, imagine the pressure exerted by a 76 cm column of mercury at sea level, at a temperature of 15 degrees Celsius.

To connect this with everyday experience, one standard atmosphere is the average pressure at sea level on Earth. Previously, pressure was often measured in millimeters of mercury (mmHg), but in scientific contexts, the pascal has become the standard unit. The importance of having a consistent reference, like the standard atmosphere, cannot be overstated as it allows scientists and engineers to compare and convert measurements of pressure regardless of the system used.

When it comes to pressure unit conversion, the process involves using known conversion factors to relate one unit of pressure to another. For example, to convert atmospheric pressure from pascals to pounds per square inch (psi), you need to know the equivalent values: 1 atm is 101325 Pa, and 1 psi is 6894.76 N/m².

With these values, a simple multiplication \(1 \text{ atm} = 101325 \text{ Pa} \times (1 \text{ psi} / 6894.76 \text{ Pa})\) gives the pressure in psi. Careful unit conversion is crucial in scientific calculations and engineering design to ensure accuracy and coherence across various systems and scales. As many equations in physics, such as the ideal gas law, are sensitive to units, mastering conversion is key to arriving at correct solutions.

When calculating mass from a known volume and density, the relationship \(\text{mass} = \text{density} \times \text{volume}\) is used. Density represents how much mass is contained within a unit volume of a material and is often expressed in grams per cubic centimeter (g/cm³) or grams per milliliter (g/mL).

For example, in the given exercise, we have a column of water where its density is 1.00 g/mL. The mass of this water can be determined by first calculating the volume of the water column in milliliters (which is numerically equivalent to cubic centimeters). Once the volume is found, multiplying by the density gives the mass in grams. Ultimately, to find the mass in pounds, a conversion is done using the conversion factor of one pound equaling 454 grams. These steps illustrate how interrelated and essential it is to understand the concepts of density and mass calculation in solving a wide array of practical problems.