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Chapter 18: Problem 23

A small electric immersion heater is used to heat \(100 \mathrm{~g}\) of water for a cup of instant coffee. The heater is labeled "200 watts" (it converts electrical energy to thermal energy at this rate). Calculate the time required to bring all this water from \(23.0^{\circ} \mathrm{C}\) to \(100^{\circ} \mathrm{C},\) ignoring any heat losses.

Short Answer

Expert verified
It takes approximately 161 seconds to heat the water.

Step by step solution

01

Calculate the Energy Needed

We first need to calculate the energy required to heat the water from 23.0掳C to 100掳C. The specific heat capacity of water is approximately 4.18 J/g掳C. The mass of the water is 100 g. The temperature change (螖T) is 100掳C - 23掳C = 77掳C. Using the formula \( q = mc\Delta T \) where \( q \) is the heat energy, \( m \) is the mass, \( c \) is the specific heat, and \( \Delta T \) is the temperature change, the energy required is \( q = 100 \times 4.18 \times 77 = 32186 \text{ J} \).
02

Determine the Power Output

The heater is labeled 200 watts. This means it provides energy at the rate of 200 joules per second.
03

Calculate the Time Required

To determine the time required to heat the water, we use the formula \( t = \frac{q}{P} \), where \( t \) is the time in seconds, \( q \) is the energy required, and \( P \) is the power output. Plugging in the known values, \( t = \frac{32186}{200} \approx 160.93 \text{ seconds} \).

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Key Concepts

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

Specific Heat Capacity
When it comes to calculating how much heat energy is needed to change the temperature of a substance, specific heat capacity comes into play. Specific heat capacity, often denoted as \( c \), is a property of a material and is defined as the amount of heat required to raise the temperature of 1 gram of the substance by 1 degree Celsius (or Kelvin). For example, the specific heat capacity of water is approximately 4.18 J/g掳C, which means it takes 4.18 Joules to heat 1 gram of water by 1掳C.
  • Importance: Specific heat capacity helps us understand how different materials respond to heat. Substances with high specific heat capacities require more energy to change their temperature compared to those with lower capacities.
  • Application: In heating processes, knowing the specific heat capacity of a substance allows us to precisely calculate the amount of energy needed to achieve a desired temperature change. In the given exercise, this concept helps us determine how much energy (32186 Joules) is needed to heat 100 g of water by 77掳C.
Energy Conversion
Energy conversion is a crucial process that involves changing energy from one form to another. In the exercise problem, we see electrical energy being converted into thermal energy through a heater. The heater's watts, or power rating, indicate how much electrical energy it can convert per unit of time.
  • Types of energy: Energy comes in various forms such as electrical, thermal, chemical, mechanical, etc. Each type can be converted into others under certain conditions.
  • Rate of conversion: The rate at which energy is converted is measured in watts (Joules per second). A 200-watt heater, for example, converts 200 Joules of electrical energy into heat every second.
  • Significance: Understanding how energy conversion works allows us to predict how long a device will take to perform a specific task. In our problem, knowing the heater's power output lets us calculate the time required to heat the water.
Problem Solving in Physics
In physics, problem solving often requires a combination of understanding concepts, applying formulas, and logical thinking. The exercise provides an excellent example of this process, involving energy, heat capacity, and time.
  • Identify the problem: Begin by understanding what is being asked. Here, it's the time needed to heat water using a specific heater.
  • Apply relevant formulas: Use known formulas to relate concepts. We used \( q = mc\Delta T \) to find the energy required and \( t = \frac{q}{P} \) to calculate the time.
  • Logical reasoning: Ensure each step logically follows from the preceding one and relate all findings back to the original question.
  • Final check: Validate the solution by checking units and seeing if the result is reasonable.
Problems like this one demonstrate the interconnectedness of concepts and the importance of a structured approach to understand complex situations.

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Most popular questions from this chapter

What mass of butter, which has a usable energy content of \(6.0 \mathrm{Cal} / \mathrm{g}(=6000 \mathrm{cal} / \mathrm{g}),\) would be equivalent to the change in gravitational potential energy of a \(73.0 \mathrm{~kg}\) man who ascends from sea level to the top of Mt. Everest, at elevation \(8.84 \mathrm{~km}\) ? Assume that the average \(g\) for the ascent is \(9.80 \mathrm{~m} / \mathrm{s}^{2}\).

Calculate the specific heat of a metal from the following data. A container made of the metal has a mass of \(3.6 \mathrm{~kg}\) and contains \(14 \mathrm{~kg}\) of water. \(\mathrm{A} 1.8 \mathrm{~kg}\) piece of the metal initially at a temperature of \(180^{\circ} \mathrm{C}\) is dropped into the water. The container and water initially have a temperature of \(16.0^{\circ} \mathrm{C}\), and the final temperature of the entire (insulated) system is \(18.0^{\circ} \mathrm{C}\).

A \(2.50 \mathrm{~kg}\) lump of aluminum is heated to \(92.0^{\circ} \mathrm{C}\) and then dropped into \(8.00 \mathrm{~kg}\) of water at \(5.00^{\circ} \mathrm{C}\). Assuming that the lumpwater system is thermally isolated, what is the system's equilibrium temperature?

A recruit can join the semi-secret "300 F" club at the Amundsen-Scott South Pole Station only when the outside temperature is below \(-70^{\circ} \mathrm{C}\). On such a day, the recruit first basks in a hot sauna and then runs outside wearing only shoes. (This is, of course, extremely dangerous, but the rite is effectively a protest against the constant danger of the cold. Assume that upon stepping out of the sauna, the recruit's skin temperature is \(102^{\circ} \mathrm{F}\) and the walls, ceiling, and floor of the sauna room have a temperature of \(30^{\circ} \mathrm{C}\). Estimate the recruit's surface area, and take the skin emissivity to be \(0.80 .\) (a) What is the approximate net rate \(P_{\text {net }}\) at which the recruit loses energy via thermal radiation exchanges with the room? Next, assume that when outdoors, half the recruit's surface area exchanges thermal radiation with the sky at a temperature of \(-25^{\circ} \mathrm{C}\) and the other half exchanges thermal radiation with the snow and ground at a temperature of \(-80^{\circ} \mathrm{C}\). What is the approximate net rate at which the recruit loses energy via thermal radiation exchanges with (b) the sky and (c) the snow and ground?

In the extrusion of cold chocolate from a tube, work is done on the chocolate by the pressure applied by a ram forcing the chocolate through the tube. The work per unit mass of extruded chocolate is equal to \(p / \rho,\) where \(p\) is the difference between the applied pressure and the pressure where the chocolate emerges from the tube, and \(\rho\) is the density of the chocolate. Rather than increasing the temperature of the chocolate, this work melts cocoa fats in the chocolate. These fats have a heat of fusion of \(150 \mathrm{~kJ} / \mathrm{kg} .\) Assume that all of the work goes into that melting and that these fats make up \(30 \%\) of the chocolate's mass. What percentage of the fats melt during the extrusion if \(p=5.5 \mathrm{MPa}\) and \(\rho=1200 \mathrm{~kg} / \mathrm{m}^{3} ?\)
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