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Chapter 12: Problem 32

Suppose you are selling apple cider for two dollars a gallon when the temperature is \(4.0^{\circ} \mathrm{C}\). The coefficient of volume expansion of the cider is \(280 \times 10^{-6}\left(\mathrm{C}^{\circ}\right)^{-1} .\) How much more money (in pennies) would you make per gallon by refilling the container on a day when the temperature is \(26^{\circ} \mathrm{C} ?\) Ignore the expansion of the container.

Short Answer

Expert verified
You would make about 1 penny more per gallon.

Step by step solution

01

Find the Temperature Change

Determine the change in temperature. The initial temperature \(T_i\) is \(4.0^{\circ} \mathrm{C}\) and the final temperature \(T_f\) is \(26^{\circ} \mathrm{C}\). The change in temperature is \(\Delta T = T_f - T_i = 26 - 4 = 22^{\circ} \mathrm{C}\).
02

Calculate Volume Expansion

Use the formula for volume expansion: \(\Delta V = \beta V_0 \Delta T\), where \(\beta = 280 \times 10^{-6} (\mathrm{C}^{\circ})^{-1}\) is the coefficient of volume expansion and \(V_0 = 1\) gallon is the initial volume. Substitute the values: \(\Delta V = 280 \times 10^{-6} \times 1 \times 22\).
03

Solve for Volume Change

Compute \(\Delta V\): \(\Delta V = 280 \times 10^{-6} \times 22 = 0.00616\) gallons. This is how much the volume of one gallon of cider expands with an increase in temperature from \(4.0^{\circ} \mathrm{C}\) to \(26^{\circ} \mathrm{C}\).
04

Calculate Additional Earnings

Since you sell cider at \$2.00 per gallon, calculate how much extra you earn from the expanded cider: \(\text{Additional Earnings} = 0.00616 \times 2 = 0.01232\) dollars.
05

Convert Dollars to Pennies

Convert the additional earnings from dollars to pennies by multiplying by 100 (since 1 dollar = 100 pennies): \(0.01232 \times 100 = 1.232\) pennies.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Volume Expansion
Volume expansion is a phenomenon where a substance increases its volume when the temperature rises. This is due to the kinetic energy of the molecules increasing, causing them to move further apart. In our cider example, when the temperature increased from 4掳C to 26掳C, the cider expanded in volume due to thermal expansion.
  • The formula used to calculate this is: \( \Delta V = \beta V_0 \Delta T \).
  • Here, \( \Delta V \) represents the change in volume, \( \beta \) is the coefficient of volume expansion, \( V_0 \) is the initial volume, and \( \Delta T \) is the change in temperature.
Understanding this concept allows us to predict and utilize the physical changes in materials as they experience different temperatures.
Temperature Change
Temperature change is the variation in temperature a substance experiences. It's crucial in the study of thermal expansion because it directly influences how much an object or substance will expand or contract.
  • The change in temperature is calculated using the formula: \( \Delta T = T_f - T_i \).
  • In the original problem, \( T_f \) is the final temperature (26掳C) and \( T_i \) is the initial temperature (4掳C).
Thus, the temperature change \( \Delta T \) is 22掳C. This change is significant because every degree of temperature change can contribute to a noticeable change in volume, especially with substances having a high coefficient of volume expansion.
Coefficient of Volume Expansion
The coefficient of volume expansion, denoted by \( \beta \), quantifies how a substance's volume changes with temperature changes. It is expressed in units of \( (掳C)^{-1} \), which describe the fractional change in volume per degree Celsius change in temperature.
  • In the cider example, \( \beta = 280 \times 10^{-6} (掳C)^{-1} \).
  • This indicates that for each degree Celsius the temperature increases, the cider's volume increases by a small fraction of its original volume.
Coefficients vary widely between substances, with liquids like cider typically having a higher coefficient compared to solids. Understanding \( \beta \) is key to accurately predicting volume changes.

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Most popular questions from this chapter

An aluminum can is filled to the brim with a liquid. The can and the liquid are heated so their temperatures change by the same amount. The can's initial volume at \(5^{\circ} \mathrm{C}\) is \(3.5 \times 10^{-4} \mathrm{m}^{3} .\) The coefficient of volume expansion for aluminum is \(69 \times 10^{-6}\left(\mathrm{C}^{\circ}\right)^{-1} .\) When the can and the liquid are heated to \(78^{\circ} \mathrm{C}, 3.6 \times 10^{-6} \mathrm{m}^{3}\) of liquid spills over. What is the coefficient of volume expansion of the liquid?

Two bars of identical mass are at \(25^{\circ} \mathrm{C} .\) One is made from glass and the other from another substance. The specific heat capacity of glass is \(840 \mathrm{J} /\left(\mathrm{kg} \cdot \mathrm{C}^{\circ}\right) .\) When identical amounts of heat are supplied to each, the glass bar reaches a temperature of \(88{ }^{\circ} \mathrm{C},\) while the other bar reaches \(250.0^{\circ} \mathrm{C} .\) What is the specific heat capacity of the other substance?

An ice chest at a beach party contains 12 cans of soda at \(5.0^{\circ} \mathrm{C}\). Each can of soda has a mass of \(0.35 \mathrm{kg}\) and a specific heat capacity of \(3800 \mathrm{J} /\left(\mathrm{kg} \cdot \mathrm{C}^{\circ}\right) .\) Someone adds a \(6.5-\mathrm{kg}\) watermelon at \(27^{\circ} \mathrm{C}\) to the chest. The specific heat capacity of watermelon is nearly the same as that of water. Ignore the specific heat capacity of the chest and determine the final temperature \(T\) of the soda and watermelon.

Occasionally, huge icebergs are found floating on the ocean's currents. Suppose one such iceberg is \(120 \mathrm{km}\) long, \(35 \mathrm{km}\) wide, and \(230 \mathrm{m}\) thick. (a) How much heat would be required to melt this iceberg (assumed to be at \(\left.0^{\circ} \mathrm{C}\right)\) into liquid water at \(0^{\circ} \mathrm{C}\) ? The density of ice is \(917 \mathrm{kg} / \mathrm{m}^{3} .\) (b) The annual energy consumption by the United States is about \(1.1 \times 10^{20} \mathrm{J}\). If this energy were delivered to the iceberg every year, how many years would it take before the ice melted?

What's your normal body temperature? It may not be \(98.6^{\circ} \mathrm{F}\), the often-quoted average that was determined in the nineteenth century. A more recent study has reported an average temperature of \(98.2^{\circ} \mathrm{F}\). What is the difference between these averages, expressed in Celsius degrees?
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