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Chapter 14: Problem 67

A black wood stove has a surface area of \(1.20 \mathrm{m}^{2}\) and a surface temperature of \(175^{\circ} \mathrm{C} .\) What is the net rate at which heat is radiated into the room? The room temperature is \(20^{\circ} \mathrm{C}\).

Short Answer

Expert verified
Answer: The net rate at which heat is radiated into the room is approximately 920.83 W.

Step by step solution

01

Convert temperatures to Kelvin

To use the Stefan-Boltzmann Law, we need to convert the given temperatures to Kelvin. The conversion formula is K = C + 273.15. Stove temperature in Kelvin: \(T_s = 175 + 273.15 = 448.15 \mathrm{K}\) Room temperature in Kelvin: \(T_r = 20 + 273.15 = 293.15 \mathrm{K}\)
02

Write the Stefan-Boltzmann Law for power emitted and absorbed

The Stefan-Boltzmann Law states that the power radiated per unit area by a black body is directly proportional to the fourth power of its absolute temperature. \(P = \sigma A T^4\), where \(P\) = radiated power, \(\sigma\) = Stefan-Boltzmann constant \((5.67 \times 10^{-8}\) W m\(^{-2}\) K\(^{-4})\), \(A\) = surface area, and \(T\) = absolute temperature.
03

Calculate the power emitted by the stove

Using the Stefan-Boltzmann Law, we can find the power emitted by the stove. \(P_s = \sigma A_s {T_s}^4\) \(P_s = (5.67 \times 10^{-8}\) W m\(^{-2}\) K\(^{-4})\) × \(1.20 \mathrm{m^2}\) × \((448.15 \mathrm{K})^4\) \(P_s \approx 1241.99 \mathrm{W}\)
04

Calculate the power absorbed by the room

Similarly, we can find the power absorbed by the room. \(P_r = \sigma A_s {T_r}^4\) \(P_r = (5.67 \times 10^{-8}\) W m\(^{-2}\) K\(^{-4})\) × \(1.20 \mathrm{m^2}\) × \((293.15 \mathrm{K})^4\) \(P_r \approx 321.16 \mathrm{W}\)
05

Find the net rate of heat radiation

The net rate of heat radiation is the difference between the power emitted by the stove and the power absorbed by the room. Net rate of heat radiation = \(P_s - P_r\) Net rate of heat radiation = \(1241.99 \mathrm{W} - 321.16 \mathrm{W}\) Net rate of heat radiation \(\approx 920.83 \mathrm{W}\) Thus, the net rate at which heat is radiated into the room is approximately \(920.83 \mathrm{W}\).

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Most popular questions from this chapter

A lizard of mass \(3.0 \mathrm{g}\) is warming itself in the bright sunlight. It casts a shadow of \(1.6 \mathrm{cm}^{2}\) on a piece of paper held perpendicularly to the Sun's rays. The intensity of sunlight at the Earth is \(1.4 \times 10^{3} \mathrm{W} / \mathrm{m}^{2},\) but only half of this energy penetrates the atmosphere and is absorbed by the lizard. (a) If the lizard has a specific heat of $4.2 \mathrm{J} /\left(\mathrm{g} \cdot^{\circ} \mathrm{C}\right),$ what is the rate of increase of the lizard's temperature? (b) Assuming that there is no heat loss by the lizard (to simplify), how long must the lizard lie in the Sun in order to raise its temperature by \(5.0^{\circ} \mathrm{C} ?\)
Imagine that 501 people are present in a movie theater of volume $8.00 \times 10^{3} \mathrm{m}^{3}$ that is sealed shut so no air can escape. Each person gives off heat at an average rate of \(110 \mathrm{W} .\) By how much will the temperature of the air have increased during a 2.0 -h movie? The initial pressure is \(1.01 \times 10^{5} \mathrm{Pa}\) and the initial temperature is \(20.0^{\circ} \mathrm{C} .\) Assume that all the heat output of the people goes into heating the air (a diatomic gas).
A spring of force constant \(k=8.4 \times 10^{3} \mathrm{N} / \mathrm{m}\) is compressed by \(0.10 \mathrm{m} .\) It is placed into a vessel containing $1.0 \mathrm{kg}$ of water and then released. Assuming all the energy from the spring goes into heating the water, find the change in temperature of the water.
A blacksmith heats a 0.38 -kg piece of iron to \(498^{\circ} \mathrm{C}\) in his forge. After shaping it into a decorative design, he places it into a bucket of water to cool. If the available water is at \(20.0^{\circ} \mathrm{C},\) what minimum amount of water must be in the bucket to cool the iron to \(23.0^{\circ} \mathrm{C} ?\) The water in the bucket should remain in the liquid phase.
A cylinder contains 250 L of hydrogen gas \(\left(\mathrm{H}_{2}\right)\) at \(0.0^{\circ} \mathrm{C}\) and a pressure of 10.0 atm. How much energy is required to raise the temperature of this gas to \(25.0^{\circ} \mathrm{C} ?\)
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