Problem 111 - Chemistry A Molecular Approach

Question

An ordinary gasoline can measuring \(30.0 \mathrm{~cm}\) by \(20.0 \mathrm{~cm}\) by \(15.0 \mathrm{~cm}\) is evacuated with a vacuum pump. Assuming that virtually all of the air can be removed from inside the can and that atmospheric pressure is 14.7 psi, what is the total force (in pounds) on the surface of the can? Do you think that the can could withstand the force?

Step-by-Step Solution

Answer

The total force on the surface of the empty gasoline can is 6150.23 pounds. To determine if the can could withstand this force, you would need to compare it to the material strength specifications for the can.

Step 1: Convert Atmospheric Pressure to Pascals

First, convert the atmospheric pressure given in pounds per square inch (psi) to pascals (Pa) using the conversion factor 1 psi = 6894.76 Pa. So, the atmospheric pressure in pascals is given by atmospheric pressure in psi multiplied by the conversion factor.Atmospheric pressure in pascals (Pa) = 14.7 psi * 6894.76 Pa/psi = 101433.37 Pa

Step 2: Calculate the Surface Area of the Can

Next, calculate the total surface area of the can. As the can is a rectangular prism, its surface area (SA) is the sum of the areas of the six sides. The formula for the surface area of a rectangular prism is:SA = 2lw + 2lh + 2whwhere l, w, h are the length, width, and height of the can respectively. Plugging in the values:SA = 2 * 30.0 cm * 20.0 cm + 2 * 30.0 cm * 15.0 cm + 2 * 20.0 cm * 15.0 cmSA = 1200 cm² + 900 cm² + 600 cm²SA = 2700 cm².Next, convert the surface area from square centimeters to square inches for compatibility with the pressure units. There are 2.54 cm in an inch, so there are (2.54 cm)² = 6.4516 cm² in a square inch. Thus:SA in square inches (in²) = SA in cm² / 6.4516 cm²/in²SA in square inches = 2700 cm² / 6.4516 cm²/in²SA in square inches = 418.58 in².

Step 3: Calculate Total Force on the Can

To find the total force exerted by the atmospheric pressure on the surface of the can, use the formula:Force (F) = Pressure (P) * Surface Area (A)Substituting pressure in pascals and area in square inches:Force in pounds (lb) = Atmospheric pressure in psi * Surface area in in²Force in pounds = 14.7 psi * 418.58 in²Force in pounds = 6150.23 lb.The total force on the surface of the can is therefore 6150.23 pounds.

Step 4: Assess the Can's Strength

Determine if the can could withstand the force by comparing the calculated force with the material strength specifications of the can, which are typically given by the manufacturer or determined by standards for the material. If the force exceeds what the material can typically withstand, the can may not be able to withstand the force.

Key Concepts

Pascal's Conversion

Understanding how to convert pressure units is fundamental in physics and engineering. Atmospheric pressure, commonly measured in pounds per square inch (psi), can be converted to Pascals (Pa), which is the SI unit of pressure. The conversion factor between these units is key: 1 psi is equivalent to 6894.76 Pa.

When faced with pressure in psi, like our atmospheric pressure of 14.7 psi, multiplication by the conversion factor yields the pressure in Pascals. Specifically,\[\text{Atmospheric pressure in Pascals (Pa)} = 14.7 \text{ psi} \times 6894.76 \frac{\text{Pa}}{\text{psi}} = 101433.37 \text{ Pa}\].

Always ensure that you're using the correct conversion factor so that your calculations are accurate. Converting units properly is the first step to ensuring that subsequent calculations, such as force on a surface, are correct.

Surface Area Calculation

The surface area of an object is a measure of the total area that the surface of the object occupies. For a box-shaped object like our gasoline can, calculating surface area involves summing the areas of all six sides of the rectangular prism.

The formula for this calculation is: \[SA = 2lw + 2lh + 2wh\], where \(l\) is length, \(w\) is width, and \(h\) is height. With our gasoline can dimensions, the surface area in square centimeters is worked out as follows:\[SA = 2 \times 30.0 \text{ cm} \times 20.0 \text{ cm} + 2 \times 30.0 \text{ cm} \times 15.0 \text{ cm} + 2 \times 20.0 \text{ cm} \times 15.0 \text{ cm} = 2700 \text{ cm}^2\].

To align the surface area with the pressure unit psi, conversion to square inches is necessary. For this, the factor \((2.54 \text{ cm})^2 = 6.4516 \text{ cm}^2/\text{in}^2\) is used to get the area in square inches: \[SA_{\text{in}^2} = 2700 \text{ cm}^2 / 6.4516 \frac{\text{cm}^2}{\text{in}^2}\approx 418.58 \text{ in}^2\].

This step is essential to correctly assess the total force exerted on the object when the applied pressure is identified in a different unit of area.

Force Calculation

The force exerted on an object by pressure is given by the product of the pressure and the area over which it is applied. When it comes to our evacuated gasoline can, this physical principle allows us to calculate the total force applied by atmospheric pressure.

The formula for calculating force is: \[F = P \times A\], where \(F\) is the force, \(P\) is the pressure, and \(A\) is the surface area. By applying the atmospheric pressure in psi and the converted surface area in square inches, we determine the force in pounds: \[F = 14.7 \text{ psi} \times 418.58 \text{ in}^2 = 6150.23 \text{ lb}\].

This substantial force results from the atmospheric pressure acting on every inch of the can’s surface area. To answer whether the can could withstand this force, we must know the material's strength; generally, manufacturers provide this information. If our force calculation exceeds the can's structural tolerance, the can would collapse under atmospheric pressure when evacuated. Understanding force calculation is imperative for designing containers and structures that can withstand their operating conditions.