Problem 93 - Chemistry and Chemical Reactivity

Question

A In July \(1983,\) an Air Canada Boeing 767 ran out of fuel over central Canada on a trip from Montreal to Edmonton. (The plane glided safely to a landing at an abandoned airstrip.) The pilots knew that 22,300 kg of fuel were required for the trip, and they knew that 7682 L of fuel were already in the tank. The ground crew added 4916 L of fuel, which was only about one fifth of what was required. The crew members used a factor of 1.77 for the fuel density-the problem is that 1.77 has units of pounds per liter and not kilograms per liter! What is the fuel density in units of kg/L? What mass of fuel should have been loaded? \((1 \mathrm{lb}=453.6 \mathrm{g} .)\)

Step-by-Step Solution

Answer

Density is 0.80388 kg/L; 27,734 L of fuel was needed.

Step 1: Convert Pounds to Kilograms

The given fuel density is 1.77 pounds per liter. First, convert this to kilograms per liter using the conversion factor 1 pound = 453.6 grams. Since 1000 grams = 1 kilogram, \[\text{1 pound} = 0.4536 \text{ kg}\]Thus, \[1.77 \text{ pounds/L} = 1.77 \times 0.4536 \text{ kg/L} = 0.80388 \text{ kg/L}\]

Step 2: Determine Total Fuel Needed in Liters

To find how many liters of fuel are required based on the density, we start with the required mass of fuel (22,300 kg). Use the density \(0.80388 \text{ kg/L}\) to convert this mass to liters:\[\text{Liters required} = \frac{22300 \text{ kg}}{0.80388 \text{ kg/L}} \approx 27734 \text{ L}\]

Step 3: Calculate Actual Liters of Fuel on Board

Add the initial fuel in the tank to the fuel added, both in liters:\[7682 \text{ L} + 4916 \text{ L} = 12598 \text{ L}\]

Step 4: Compare Liters of Fuel

To see if the plane had enough fuel, compare the actual liters of fuel on board (12,598 L) with the liters needed (27,734 L). It shows that they had less than half the required fuel.

Step 5: Find Actual Mass of Loaded Fuel

Calculate the actual mass of the loaded fuel using the fuel density\[\text{Mass of loaded fuel} = 12598 \text{ L} \times 0.80388 \text{ kg/L} \approx 10121.5 \text{ kg}\]

Step 6: Calculate Shortfall in Mass

Determine the shortfall by finding out how much more fuel mass was needed:\[22300 \text{ kg (required)} - 10121.5 \text{ kg (actual loaded)} \approx 12178.5 \text{ kg}\]

Key Concepts

Unit Conversion in Chemistry

Unit conversion is a crucial skill in chemistry that allows us to transform measurements from one unit of data into another. This is evident in problems involving fuel density conversion, where incorrect conversions can lead to miscalculations with serious consequences.
In the exercise, the original error was using a density factor of 1.77 in pounds per liter rather than kilograms per liter.
To fix this, we start by understanding that 1 pound is equivalent to 453.6 grams, which is then converted to kilograms. Since there are 1000 grams in a kilogram, this conversion process becomes:

• 1 lb = 0.4536 kg

To convert the fuel density from pounds per liter to kilograms per liter, multiply the given density by the appropriate conversion factor:

• 1.77 pounds/L = 1.77 × 0.4536 kg/L = 0.80388 kg/L

By mastering unit conversion, we ensure that calculations are accurate, which is essential in fields where precision is necessary, such as in aviation.

Mass and Volume Relationship

Understanding the relationship between mass and volume is essential, especially in applications involving liquids like fuel. This relationship is defined by the concept of density, a measure of mass per unit volume. Knowing either the mass or volume allows us to calculate the other, given the density.
In this exercise, the fuel density of jet fuel is crucial for solving how much additional fuel is needed for the flight. The relationship between mass (\(m\)) and volume (\(V\)) through density (\(d\)) is described by the formula:

• \(d = \frac{m}{V}\)

When we need to find out how many liters of fuel were needed, given the mass (22,300 kg) and the density (0.80388 kg/L), we use the formula in rearranged form:

• \(V = \frac{m}{d} = \frac{22300\ \text{kg}}{0.80388 \ \text{kg/L}}\)
• Which gives approximately 27,734 liters.

This calculation helps us understand how much volume a mass of a certain density will occupy, directly tying mass and volume through density.

Density Calculations

Density calculations allow us to precisely relate mass to volume. The density of a substance tells us how much mass is contained in a particular volume, which is particularly important in the context of fuel loading on airlines. Incorrect densities will lead to insufficient fuel and potentially dangerous situations.
In our scenario, the density calculation determined how much fuel was needed for the trip. With a corrected density of 0.80388 kg/L, the actual mass of the loaded fuel could also be computed:

• Calculating actual mass: 12,598 L of fuel loaded × 0.80388 kg/L = 10,121.5 kg

These calculations revealed a significant shortfall from the required 22,300 kg of fuel. Such calculations are vital in operational planning and safety management within aviation. Always ensure that density is accurately determined and utilized to avoid costly and potentially serious errors.