Problem 57 - Chemistry The Molecular Nature of Matter

Question

Suppose that you were pumping an automobile tire with a hand pump that pushed 24.0 in. \(^{3}\) of air into the tire on each stroke, and that during one such stroke the opposing pressure in the tire was \(30.0 \mathrm{lb} / \mathrm{in} .^{2}\) above the normal atmospheric pressure of \(14.7 \mathrm{lb} / \mathrm{in} .^{2} .\) Calculate the number of joules of work accomplished during each stroke. \((1 \mathrm{~L} \mathrm{~atm}=101.325 \mathrm{~J})\)

Step-by-Step Solution

Answer

The amount of work accomplished during each stroke is 97.4 J.

Step 1: Convert Pressure to Absolute Pressure

The tire has an opposing pressure of 30.0 lb/in^2 above atmospheric pressure. To find the total pressure exerted against the pump, we add this to the normal atmospheric pressure of 14.7 lb/in^2. Absolute Pressure = Opposing Pressure + Atmospheric Pressure = 30.0 lb/in^2 + 14.7 lb/in^2.

Step 2: Calculate Total Pressure in Atmospheres

First, convert the pressure in lb/in^2 to atmospheres (atm) using the fact that 1 atm equals 14.7 lb/in^2. Total Pressure (atm) = Absolute Pressure (lb/in^2)/(14.7 lb/in^2 per atm).

Step 3: Calculate Volume in Liters

Since work is given by pressure multiplied by volume change in units of L*atm, we convert the volume from cubic inches to liters. We know that 1 L = 61.024 in^3. Therefore, Volume (L) = 24.0 in^3 * (1 L/61.024 in^3).

Step 4: Calculate the Work in L*atm

Now, calculate the work done during the stroke. Work (L*atm) = Total Pressure (atm) * Volume (L).

Step 5: Convert Work from L*atm to Joules

Finally, convert the work from L*atm to joules using the conversion factor 1 L*atm = 101.325 J. Total work done (J) = Work (L*atm) * (101.325 J/L*atm). Calculate the final numerical value to obtain the work done in joules.

Key Concepts

Understanding Absolute Pressure in Physics Work Calculations

When we talk about 'absolute pressure' in physics, we're looking at the total pressure exerted on an object, which includes all the forces applied to it by the surrounding environment. In everyday terms, think of absolute pressure as the real pressure you feel when, for example, you're underwater. It combines the pressure due to the water itself and the atmosphere above it.

Absolute Pressure in Tire Pumping

In the exercise with the tire pump, absolute pressure becomes crucial to know because the pump must overcome both the tire's internal pressure and atmospheric pressure to add air. The exercise takes the pressure above the atmospheric pressure and adds it to the normal atmospheric pressure to find the absolute pressure against which the pump works. This is vital in calculating work, as work is a product of force exerted through a distance – or in the case of a gas, through the change in volume at a defined pressure.

Remembering that atmospheric pressure exists and affects all objects at the surface is important; thus, absolute pressure is always atmospheric pressure plus any additional pressure present. In other words,
Absolute Pressure = Normal Atmospheric Pressure + Extra Pressure from Other Sources.

Understanding this concept ensures the accuracy of work calculations not just in school problems but also in real-world scenarios like filling a car tire or measuring the pressure in a steam boiler.

Deciphering Pressure-Volume Work in a Physical Context

The term 'pressure-volume work' is used to describe the work done when the volume of a gas changes in the presence of an external pressure. In a more technical sense, this work can be expressed as the product of pressure (force per unit area) and the change in volume. For example, when inflating a tire, the pump does work by displacing air, changing its volume under the influence of a certain pressure.

Pressure-Volume Work in Our Tire Pump Example

In the exercise provided, we're looking at how the pump performs work by moving air into the tire. The air's movement causes a change in volume within the tire, which occurs against the opposing pressure. To calculate this type of work, we first establish the total pressure the pump is working against – this includes both atmospheric and the tire's internal pressure. Then, we look at how much volume is changing – the air being pushed in by each stroke of the pump. Multiplying these two values gives us the work done per stroke in terms of pressure-volume, which forms a foundation before we convert to energy units like joules.

Thus, in a simple expression:
Pressure-Volume Work = Pressure × Change in Volume

Applying this in practical situations requires careful measurement of pressure and volume changes, which is a fundamental concept in thermodynamics and fluid dynamics.

Navigating Unit Conversion in Physics Problems

Unit conversion is the process of changing a measure to a different unit without changing the quantity's value, which is often essential in physics to ensure that equations and calculations are consistent and meaningful. In physics problems, we frequently need to convert units so that they align with standard units of measurement, like meters instead of feet, or kilograms instead of pounds.

The Importance of Unit Conversion in Work Calculations

Let's look at the example of pumping the tire again. When you are calculating the work done, the volume of air and the pressure must be in compatible units to apply the formula correctly. Often in physics, these units need to be in the international system (SI) to ensure consistency. For example, the volume initially given in cubic inches has been converted to liters, and pressure in pounds per square inch to atmospheres, before finally converting the work done from liter-atmospheres to joules.

Unit conversion may seem like a simple mathematical step, but it's a fundamental skill in physics. It allows for:

• Direct comparison of quantities.
• Accurate scientific calculation and communication.
• Application of proper formulas that often require a specific unit measurement.

It ensures that the final result, the work done during each stroke in this case, is presented in an internationally understood and applicable unit – the joule. Grasping unit conversion means you're not just tackling a math problem, but you're also preparing to communicate and work within the global scientific community.