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Chapter 13: Problem 24

What energy is required to raise the temperature of \(450 \mathrm{~g}\) of water by \(6^{\circ} \mathrm{C} ?\) (a) \(4200 \mathrm{~J}\); (b) \(5200 \mathrm{~kJ} ;\) (c) \(6600 \mathrm{~kJ}\); (d) \(11,300 \mathrm{~J}\).

Short Answer

Expert verified
The energy required is approx. \( 11,300 \text{ J} \) (option d).

Step by step solution

01

Understand the Formula

To calculate the energy required to raise the temperature, we use the formula \( Q = mc\Delta T \), where \( Q \) is the heat energy, \( m \) is the mass, \( c \) is the specific heat capacity, and \( \Delta T \) is the change in temperature. For water, the specific heat capacity \( c \) is \( 4.18 \text{ J/g掳C} \).
02

Identify Given Values

From the problem, we know the mass \( m = 450 \text{ g} \) and the temperature change \( \Delta T = 6掳C \).
03

Apply the Formula

Substitute the known values into the formula: \( Q = (450 \text{ g}) \times (4.18 \text{ J/g掳C}) \times (6掳C) \).
04

Calculate the Energy

Perform the multiplication to find \( Q \):\[Q = 450 \times 4.18 \times 6 = 11286 \text{ J}\]
05

Select the Best Option

Compare the calculated energy value with the options provided in the question. The closest option to \( 11286 \text{ J} \) is \( 11,300 \text{ J} \).

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Key Concepts

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

Specific Heat Capacity
Specific heat capacity is a fundamental property of materials that tells us how much energy is needed to change the temperature of a given mass of a substance by one degree Celsius. For instance, water has a specific heat capacity of approximately 4.18 J/g掳C.

  • This means you need 4.18 joules of energy to raise the temperature of 1 gram of water by 1掳C.
  • The higher the specific heat capacity, the more energy is required to change the temperature.
  • Specific heat plays a crucial role in many real-world applications, such as climate modeling and cooking.
  • It helps us understand why it takes longer to heat water compared to substances with lower specific heat capacities.
Understanding specific heat capacity empowers you to make accurate predictions about how a substance will react to heat energy.
Temperature Change
Temperature change is a key factor when calculating the amount of heat energy needed in a process. It is simply the difference between the final and initial temperatures. In our exercise, the water's temperature is raised by 6掳C.

  • It is represented by the symbol \( \Delta T \), where \( \Delta \) indicates a change and \( T \) stands for temperature.
  • A positive temperature change means an increase in temperature, while a negative change indicates cooling.
  • Understanding temperature change is essential for many scientific and engineering applications, such as energy management and materials science.
By carefully measuring temperature changes, we can calculate the energy required for heating or cooling processes accurately.
Energy Formula
The energy formula \( Q = mc\Delta T \) is vital in physics for calculating how much energy is required to bring about a temperature change in a substance.

  • Here, \( Q \) stands for the heat energy absorbed or released, measured in joules.
  • \( m \) represents the mass of the substance, which is crucial as it scales the energy requirement proportionally.
  • \( c \) is the specific heat capacity, a constant that varies with different substances.
  • \( \Delta T \) represents the change in temperature, a crucial parameter in evaluating energy needs.
This formula seamlessly combines these quantities, allowing you to solve practical problems related to heat energy management in various fields such as chemistry, engineering, and environmental science.
Physics Problem Solving
Solving physics problems, such as calculating heat energy, involves systematic approaches to ensure accuracy and comprehension.

  • First, understand the problem and identify the known and unknown values.
  • Apply relevant formulas, like \( Q = mc\Delta T \), replacing variables with the given values.
  • Perform calculations carefully, double-checking each step.
  • Compare your result with provided options or benchmark values for validation.
  • Finally, reflect on the solution to ensure it aligns with physical intuition, such as whether the calculated energy seems reasonable given the circumstances.
Developing strong problem-solving skills in physics will empower you to tackle various scientific challenges with confidence and clarity. Thorough practice leads to greater ease in interpreting and applying concepts effectively.

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