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Chapter 6: Problem 63

You wish to heat water to make coffee. How much heat (in joules) must be used to raise the temperature of \(0.180 \mathrm{~kg}\) of tap water (enough for one cup of coffee) from \(19^{\circ} \mathrm{C}\) to \(96^{\circ} \mathrm{C}\) (near the ideal brewing temperature)? Assume the specific heat is that of pure water, \(4.18 \mathrm{~J} /\left(\mathrm{g} \cdot{ }^{\circ} \mathrm{C}\right)\).

Short Answer

Expert verified
The heat required is approximately 58045.2 joules.

Step by step solution

01

Identify the Given Parameters

In this problem, we are given the mass of water, which is \(0.180 \text{ kg}\) or \(180 \text{ g}\) since 1 kg = 1000 g. The initial temperature \(T_i\) is \(19^{\circ} \text{C}\) and the final temperature \(T_f\) is \(96^{\circ} \text{C}\). The specific heat capacity of water is given as \(4.18 \text{ J/g掳C}\).
02

Calculate the Temperature Change

To find the heat required, we need to calculate the change in temperature: \(\Delta T = T_f - T_i\). Substitute the given values: \(\Delta T = 96^{\circ} \text{C} - 19^{\circ} \text{C} = 77^{\circ} \text{C}\).
03

Apply the Heat Equation

The heat needed can be calculated using the formula: \( Q = mc\Delta T \) where \(m\) is the mass of the water in grams, \(c\) is the specific heat capacity, and \(\Delta T\) is the temperature change. Substitute the known values into the formula: \( Q = 180 \text{ g} \times 4.18 \text{ J/g掳C} \times 77^{\circ} \text{C} \).
04

Perform the Calculation

Now calculate the heat \(Q\):\[ Q = 180 \times 4.18 \times 77 \] \[ Q = 58045.2 \text{ J} \]Therefore, the amount of heat needed is \(58045.2 \text{ joules}\).

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Key Concepts

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

Specific Heat Capacity
When heating substances, knowing the specific heat capacity is essential. It defines how much heat energy is needed to raise the temperature of one gram of a substance by one degree Celsius. In our coffee example, we used pure water, which has a specific heat capacity of \(4.18 \text{ J/g掳C}\). This means every gram of water requires \(4.18\) joules to raise its temperature by \(1^{\circ} \text{C}\). This constant is crucial in determining how much total energy is needed.

Some points to understand about specific heat capacity are:
  • It varies between different substances based on their intermolecular forces.
  • Substances with higher specific heat capacities can store more heat without a large change in temperature.
  • A specific heat capacity is usually given in units of \(\text{J/g掳C}\) or \(\text{J/kg掳C}\), so it's important to match these units with your mass measurements.
Temperature Change
Understanding temperature change is crucial when calculating the heat required for processes like making coffee. The temperature change (\(\Delta T\)) represents the difference between the final and initial temperatures of a substance. For our problem, the initial temperature of the water was \(19^{\circ} \text{C}\) and the final was \(96^{\circ} \text{C}\). By subtracting the initial from the final temperature, we found the water needed to change \(77^{\circ} \text{C}\) to reach the ideal coffee temperature.

Here's what you need to keep in mind about temperature change:
  • Temperature change is calculated as \(\Delta T = T_f - T_i\), where \(T_f\) is the final temperature and \(T_i\) is the initial temperature.
  • The result of \(\Delta T\) can be positive or negative depending on whether the substance is being heated or cooled.
  • Accurate measurement of \(\Delta T\) is paramount for precise heat calculations.
Joules
Joules serve as the unit of measurement for energy, specifically in the context of heat energy in this exercise. When discussing how much energy is required to heat water, we're talking about how many joules are needed to change the temperature. In our exercise, we found \(58045.2\) joules were necessary to heat the water to the ideal brewing temperature.

Understanding joules is key in thermodynamics and energy calculations. Consider these important facts:
  • 1 Joule is equivalent to the energy transferred when a force of one newton moves an object one meter.
  • In terms of heat, joules measure the energy needed to increase the temperature of substances.
  • This measurement helps in comparing the energy usage and efficiency of different heating methods and substances.
This implies a real-world application by determining how much energy you need to use in various daily tasks.

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Most popular questions from this chapter

Part 1: In an insulated container, you mix 200. g of water at \(80^{\circ} \mathrm{C}\) with \(100 . \mathrm{g}\) of water at \(20^{\circ} \mathrm{C}\). After mixing, the temperature of the water is \(60^{\circ} \mathrm{C}\). a. How much did the temperature of the hot water change? How much did the temperature of the cold water change? Compare the magnitudes (positive values) of these changes. b. During the mixing, how did the heat transfer occur: from hot water to cold, or from cold water to hot? C. What quantity of heat was transferred from one sample to the other? d. How does the quantity of heat transferred to or from the hot-water sample compare with the quantity of heat transferred to or from the cold-water sample? e. Knowing these relative quantities of heat, why is the temperature change of the cold water greater than the magnitude of the temperature change of the hot water. f. A sample of hot water is mixed with a sample of cold water that has twice its mass. Predict the temperature change of each of the samples. g. You mix two samples of water, and one increases by \(20^{\circ} \mathrm{C}\), while the other drops by \(60^{\circ} \mathrm{C}\). Which of the samples has less mass? How do the masses of the two water samples compare? h. A 7 -g sample of hot water is mixed with a \(3-\mathrm{g}\) sample of cold water. How do the temperature changes of the two water samples compare? Part \(2:\) A sample of water is heated from \(10^{\circ} \mathrm{C}\) to \(50^{\circ} \mathrm{C}\). Can you calculate the amount of heat added to the water sample that caused this temperature change? If not, what information do you need to perform this calculation? Part 3: Two samples of water are heated from \(20^{\circ} \mathrm{C}\) to \(60^{\circ} \mathrm{C}\). One of the samples requires twice as much heat to bring about this temperature change as the other. How do the masses of the two water samples compare? Explain your reasoning.

Consider the Haber process: $$ \mathrm{N}_{2}(g)+3 \mathrm{H}_{2}(g) \longrightarrow 2 \mathrm{NH}_{3}(g) ; \Delta H^{\circ}=-91.8 \mathrm{~kJ} $$ The density of ammonia at \(25^{\circ} \mathrm{C}\) and \(1.00 \mathrm{~atm}\) is \(0.696 \mathrm{~g} / \mathrm{L}\). The density of nitrogen, \(\mathrm{N}_{2}\), is \(1.145 \mathrm{~g} / \mathrm{L}\), and the molar heat capacity is \(29.12 \mathrm{~J} /\left(\mathrm{mol} \cdot{ }^{\circ} \mathrm{C}\right)\). (a) How much heat is evolved in the production of \(1.00 \mathrm{~L}\) of ammonia at \(25^{\circ} \mathrm{C}\) and \(1.00 \mathrm{~atm} ?\) (b) What percentage of this heat is required to heat the nitrogen required for this reaction \((0.500 \mathrm{~L})\) from \(25^{\circ} \mathrm{C}\) to \(400^{\circ} \mathrm{C}\), the temperature at which the Haber process is run?

A 29.1-mL sample of \(1.05 \mathrm{M}\) KOH is mixed with \(20.9 \mathrm{~mL}\) of \(1.07 M \mathrm{HBr}\) in a coffee-cup calorimeter (see Section \(6.6\) of your text for a description of a coffee-cup calorimeter). The enthalpy of the reaction, written with the lowest wholenumber coefficients, is \(-55.8 \mathrm{~kJ} .\) Both solutions are at \(21.8^{\circ} \mathrm{C}\) prior to mixing and reacting. What is the final temperature of the reaction mixture? When solving this problem, assume that no heat is lost from the calorimeter to the surroundings, the density of all solutions is \(1.00 \mathrm{~g} / \mathrm{mL}\), and volumes are additive.

Nitrous oxide, \(\mathrm{N}_{2} \mathrm{O}\), has been used as a dental anesthetic. The average speed of an \(\mathrm{N}_{2} \mathrm{O}\) molecule at \(25^{\circ} \mathrm{C}\) is \(379 \mathrm{~m} / \mathrm{s}\). Calculate the kinetic energy (in joules) of an \(\mathrm{N}_{2} \mathrm{O}\) molecule traveling at this speed.

A \(250-\mathrm{g}\) sample of water at \(20.0^{\circ} \mathrm{C}\) is placed in a freezer that is held at a constant temperature of \(-20.0^{\circ} \mathrm{C}\). Considering the water as the "system," answer the following questions: a. what is the sign of \(q_{\text {sys }}\) for the water after it is placed in the freezer? b. After a few hours, what will be the state of the water? c. How will the initial enthalpy for the water compare with the final enthalpy of the water after it has spent several hours in the freezer? d. What will the temperature of the water be after several hours in the freezer?
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