Problem 15 - Chemistry: The Central Science

Question

Suppose that a woman weighing \(130 \mathrm{lb}\) and wearing highheeled shoes momentarily places all her weight on the heel of one foot. If the area of the heel is \(0.50 \mathrm{in}^{2}\), calculate the pressure exerted on the underlying surface in (a) kilopascals, (b) atmospheres, and (c) pounds per square inch.

Step-by-Step Solution

Answer

The pressure exerted on the surface by the woman's heel is approximately (a) 1793 kilopascals (kPa), (b) 17.69 atmospheres (atm), and (c) 260 pounds per square inch (psi).

Step 1: Calculate the pressure in pounds per square inch (psi)

Use the formula for pressure to find the pressure exerted on the surface: Pressure (P) = Force (F) / Area (A) The force comes from the woman's weight, which is given as 130 lb. The area of the heel is given as 0.50 in². Plug in their values: P = 130 lb / 0.50 in² = 260 psi

Step 2: Convert the pressure to kilopascals (kPa)

To convert the pressure from psi to kPa, we can use the following conversion factor: 1 psi ≈ 6.895 kPa Now, multiply the pressure in psi by the conversion factor: 260 psi × 6.895 kPa/psi ≈ 1793 kPa

Step 3: Convert the pressure to atmospheres (atm)

To convert the pressure from psi to atm, we can use the following conversion factor: 1 psi ≈ 0.06804 atm Now, multiply the pressure in psi by the conversion factor: 260 psi × 0.06804 atm/psi ≈ 17.69 atm #Conclusion# The pressure exerted on the surface by the woman's heel is approximately: (a) 1793 kilopascals (kPa) (b) 17.69 atmospheres (atm) (c) 260 pounds per square inch (psi)

Key Concepts

Understanding Kilopascals

Pressure is a measure of force distributed over an area, and the kilopascal (kPa) is one common unit used to express pressure. One Pascal (Pa) is equal to one Newton per square meter, and a kilopascal is just 1000 Pascals. This means that it simplifies the expression of larger pressure values, as kilopascals are used for practical applications where values might otherwise become large. For example, atmospheric pressure at sea level is about 101.3 kPa.

• The prefix 'kilo-' signifies a thousand, so a conversion from Pascals to kilopascals requires dividing by 1000.
• In the exercise example, converting from pounds per square inch to kilopascals involves using the factor 1 psi ≈ 6.895 kPa, directly multiplying to yield approximately 1793 kPa.

The kilopascal is a very versatile unit used in various fields like engineering and meteorology, making it handy for students to get comfortable with.

Understanding Atmospheres

The atmosphere (atm) is a unit of measurement that has significance mostly in the context of atmospheric pressure. Pressure that we naturally encounter in daily life, especially at sea level, is termed 1 atmosphere. This metric is beneficial because it normalizes the pressure related to the ambient air that surrounds us.

• An atmosphere is defined as 101,325 Pascals, and in conversion from psi, 1 psi ≈ 0.06804 atm.
• In our exercise, this conversion results in an approximate value of 17.69 atmospheres, when applying the pressure exertion of the heel.

Using atmospheres gives individuals a more apparent intuitive grasp of large pressures considering it relates to the air pressurization around us.

Understanding Pounds per Square Inch

Pounds per square inch (psi) is a common unit of pressure primarily used in the United States. It relates the amount of force (in pounds) applied to one square inch of area. This unit is quite intuitive in everyday physical scenarios, such as measuring tire pressure or even the force exerted by an object on a surface.

• In the exercise, the woman's weight is 130 pounds applied over the heel's area of 0.5 square inches, calculated straightforwardly as 260 psi.
• Converting to other units, like kilopascals or atmospheres, often involves standard factors - showing how versatile psi can be across different systems.

By understanding psi, students can relate this information to many practical, real-world applications, from engineering to day-to-day activities.