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Chapter 3: Problem 87

If it takes \(103 \mathrm{J}\) of energy to warm a certain mass of iron from \(25^{\circ} \mathrm{C}\) to \(50 .^{\circ} \mathrm{C},\) then it will take ____ J to warm the same mass of iron from \(25^{\circ} \mathrm{C}\) to \(75^{\circ} \mathrm{C}\).

Short Answer

Expert verified
It will take \(206 \mathrm{J}\) of energy to warm the same mass of iron from \(25^{\circ} \mathrm{C}\) to \(75^{\circ}\mathrm{C}\).

Step by step solution

01

Identify the known variables and formula

We know the following variables: - The initial energy E1 = 103 J - The initial change in temperature 螖T1 = 50掳C - 25掳C = 25掳C - The final change in temperature 螖T2 = 75掳C - 25掳C = 50掳C We will use the formula E = mc螖T.
02

Calculate the ratio of 螖T2 to 螖T1

Since we're given the energy required for the initial change in temperature and want to find the energy required for the final change in temperature, we can find the ratio of 螖T2 to 螖T1: \( \frac{\Delta T_2}{\Delta T_1} = \frac{50^\circ \mathrm{C}}{25^\circ \mathrm{C}} = 2 \)
03

Find the energy required for 螖T2

Since the mass and specific heat capacity remains constant, the energy required will be directly proportional to the change in temperature. Therefore, we can multiply the initial energy by the ratio of 螖T2 to 螖T1 to find the energy required for 螖T2: E2 = E1 脳 2 = 103 J 脳 2 = 206 J Hence, it will take 206 J of energy to warm the same mass of iron from 25掳C to 75掳C.

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Key Concepts

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

Energy Calculation
When dealing with energy calculations involving heating substances, one often uses the specific heat capacity formula, which is expressed as:\[ E = mc\Delta T \]Here's a breakdown of each component:
  • \( E \) is the energy required to change the temperature, measured in joules (J).
  • \( m \) is the mass of the substance being heated, typically in kilograms (kg).
  • \( c \) is the specific heat capacity, which indicates how much energy is needed to raise the temperature of 1 kg of the substance by 1掳C, and is measured in J/(kg掳C).
  • \( \Delta T \) is the change in temperature, expressed in degrees Celsius (掳C).
In our example, the mass and the specific heat capacity of iron are not explicitly given, but they remain unchanged as the temperature increases. Therefore, when the change in temperature doubles (from 25掳C to 50掳C and then from 25掳C to 75掳C), the energy needed doubles as well. Thus, if it takes 103 J to increase the temperature by 25掳C, it will require 206 J to increase it by 50掳C.
Temperature Change
Temperature change is the difference between the final and initial temperatures, represented as \( \Delta T \). In processes like heating, this indicates how much the temperature rises. The formula for change in temperature is:\[ \Delta T = T_{\text{final}} - T_{\text{initial}} \]For the given problem, we have:
  • The initial change in temperature \( \Delta T_1 \) is from 25掳C to 50掳C, resulting in a \( \Delta T \) of 25掳C.
  • The final change in temperature \( \Delta T_2 \) extends from 25掳C to 75掳C, giving a \( \Delta T \) of 50掳C.
Understanding how temperature change relates to energy usage is crucial. As seen, the greater the change in temperature, the more energy is required, assuming mass and specific heat capacity are constant. This is why a doubling in the temperature change results in an energy doubling in our scenario.
Iron Properties
Iron is a common metal used in various applications, and understanding its thermal properties is essential for energy calculations. One key property is its specific heat capacity, typically about 0.449 J/(g掳C) or 449 J/(kg掳C). This means that raising the temperature of 1 kg of iron by 1掳C requires 449 J of energy. Here are some quick facts about iron's properties in thermal contexts:
  • Iron's specific heat capacity is relatively low compared to water, meaning it heats up quickly.
  • In the given problem, even without directly knowing the mass or specific heat capacity, we can infer behavior through the formula \( E = mc\Delta T \).
  • Keeping the mass of iron constant means any energy changes stem from differences in the temperature change, as seen in our scenario.
Understanding these aspects of iron helps in predicting how it will respond to heating and incorporating it into broader energy-related calculations.

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Most popular questions from this chapter

Describe how the process of filtration could be used to separate a mixture into its components. Give an example.

Classify the following as physical or chemical changes/ properties. a. A shirt scorches when you leave the iron on one spot too long. b. The tires on your car seem to be getting flat in very cold weather. c. Your grandmother's silver tea set gets black with tarnish over time. d. A bottle of wine left open turns to vinegar. e. Spray-on oven cleaner converts grease in the oven into a soapy material. f. An ordinary flashlight battery begins to leak with age and can't be recharged. g. Acids produced by bacteria in plaque cause teeth to decay. h. Sugar will char if overheated while making homemade candy. i. Hydrogen peroxide fizzes when applied to a wound. j. Dry ice "evaporates" without melting as time passes. k. Chlorine laundry bleaches will sometimes change the color of brightly colored clothing.

Perform the following conversions. a. \(5.442 \times 10^{4} \mathrm{J}\) to kilojoules b. \(5.442 \times 10^{4} \mathrm{J}\) to calories c. 352.6 kcal to kilojoules d. \(17.24 \mathrm{kJ}\) to kilocalories

Solutions containing copper(II) ions are bright blue in color. When sodium hydroxide is added to such a solution, a solid material forms that is colored a much paler shade of blue than the original solution of copper(II) ions. The fact that a solution containing copper(II) ions is bright blue is a ___ property.

Aqueous solutions of the substance nickel(II) sulfate are bright green in color. If an aqueous solution of barium chloride is added to an aqueous solution of nickel(II) sulfate, a white precipitate of barium sulfate forms. From the information above, indicate one physical property of nickel(II) sulfate in solution.
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