Question

When \(1.0 \mathrm{~g}\) of gasoline burns, it releases 11 kcal of heat. The density of gasoline is \(0.74 \mathrm{~g} / \mathrm{mL}\). a. How many megajoules are released when \(1.0\) gal of gasoline burns? b. If a television requires \(150 \mathrm{~kJ} / \mathrm{h}\) to run, how many hours can the television run on the energy provided by \(1.0\) gal of gasoline?

Short answer

a) 128.98 MJb) 859.87 hours

Step-by-step solution

- Convert gallons to grams

First, we need to convert the volume of gasoline from gallons to milliliters. Knowing that 1 gallon is equivalent to 3,785.41 milliliters (mL), we can write:1 gal = 3,785.41 mL

- Convert milliliters to grams

Use the density of gasoline to convert the volume to mass. Density is given as 0.74 g/mL. Therefore:Mass = Volume × DensityMass = 3,785.41 mL × 0.74 g/mL = 2,800.19 g

- Calculate total heat energy in kcal

The problem states that burning 1.0 g of gasoline releases 11 kcal of heat. Thus, for 2,800.19 g of gasoline, the total heat energy is:Energy = 2,800.19 g × 11 kcal/g = 30,802.09 kcal

- Convert kcal to megajoules (MJ)

To convert kcal to MJ, use the conversion factor: 1 kcal = 0.004184 MJEnergy = 30,802.09 kcal × 0.004184 MJ/kcal = 128.98 MJHence, burning 1.0 gal of gasoline releases 128.98 MJ.

- Identify energy consumption of television

Next, we need to find the running time of a television that requires 150 kJ/h. First, convert the energy obtained from gasoline (in MJ) to kJ, knowing that 1 MJ = 1,000 kJ:128.98 MJ = 128,980 kJ

- Calculate running time of the television

Determine how many hours the television can run by dividing the total energy obtained (in kJ) by the energy required per hour:Hours = Total Energy / Energy Consumption per HourHours = 128,980 kJ / 150 kJ/h ≈ 859.87 hours

Key concepts

Density and Volume Conversion

In this exercise, we need to convert the volume of gasoline from gallons to grams to find the total heat energy produced. To begin, we should convert the volume from gallons to milliliters. We know that 1 gallon equals 3,785.41 milliliters.
Understanding density is crucial because it helps us convert volume into mass. Density is defined as mass per unit volume, commonly represented by the formula: \( \text{Density} = \frac{\text{Mass}}{\text{Volume}} \). Since the density of gasoline is given as 0.74 g/mL, using the formula, we can find the mass when the volume is known. For example:

• Given: 1 gallon = 3,785.41 mL
• Density of gasoline: 0.74 g/mL
• We calculate mass as: \(\text{Mass} = 3,785.41 \text{mL} \times 0.74 \text{g/mL} = 2,800.19 \text{g}\)

With these conversions, the problem's volume in gallons is now converted into mass in grams, which is essential for our next calculations.

Heat Energy Calculation

Upon converting the volume to mass, we can calculate the total heat energy produced. We know that burning 1.0 g of gasoline releases 11 kcal of heat. For the mass of 2,800.19 g, the total amount of heat energy released can be calculated as:
\( \text{Total Heat Energy} = 2,800.19 \text{g} \times 11 \text{kcal/g} = 30,802.09 \text{kcal} \)
This step demonstrates a direct proportion where a given amount of mass corresponds to a certain amount of energy released.

Unit Conversion (kcal to MJ)

Now, to convert the calculated heat energy from kilocalories (kcal) to megajoules (MJ), we use a conversion factor. We know the conversion factor is:

• 1 kcal = 0.004184 MJ

So to convert the total heat energy, we use:

• Total Heat Energy in MJ = 30,802.09 kcal × 0.004184 MJ/kcal = 128.98 MJ

Making such unit conversions is critical because energy consumption in different contexts, such as electronic appliances, is usually measured in joules or kilojoules rather than calories.

Energy Consumption

Finally, we want to determine how long a television can run on the energy provided by the 1 gallon of gasoline. We start by converting the total heat energy in MJ to kJ, knowing that 1 MJ = 1,000 kJ.

• 128.98 MJ = 128,980 kJ

Next, using the given energy consumption rate of the television, which is 150 kJ/h, we can find out how many hours the television can run. The calculation is:

• Hours = Total Energy / Energy Consumption per Hour
• Hours = 128,980 kJ / 150 kJ/h ≈ 859.87 hours

Thus, the television can run for approximately 860 hours on the energy provided by burning 1 gallon of gasoline. This highlights how understanding and calculating energy transformations can be applied to real-world scenarios such as energy usage in households.