A ceramic teapot (e = 0.70) and a shiny metal one (e = 0.10) each hold 0.55 L of tea at 85°C. (a) Estimate the rate of heat loss from each, and (b) estimate the temperature drop after 30 min for each. Consider only radiation, and assume the surroundings are at 20°C.

검증된 단계별 안내

Step 1: Understand the problem. The problem involves heat loss due to radiation, which is governed by the Stefan-Boltzmann law: P = eσA(T⁴ - Tₛ⁴), where P is the power (rate of heat loss), e is the emissivity, σ is the Stefan-Boltzmann constant (5.67 × 10⁻⁸ W/m²·K⁴), A is the surface area, T is the temperature of the object in Kelvin, and Tₛ is the temperature of the surroundings in Kelvin.

Step 2: Convert the given temperatures to Kelvin. Use the formula T(K) = T(°C) + 273.15. For the tea, T = 85 + 273.15, and for the surroundings, Tₛ = 20 + 273.15.

Step 3: Estimate the surface area of the teapots. Assume the teapots are approximately spherical. The volume of tea is given as 0.55 L (or 0.55 × 10⁻³ m³). For a sphere, the volume is V = (4/3)πr³, and the surface area is A = 4πr². Solve for the radius r from the volume, then calculate the surface area A.

Step 4: Calculate the rate of heat loss (P) for each teapot using the Stefan-Boltzmann law. Substitute the emissivity (e = 0.70 for the ceramic teapot and e = 0.10 for the shiny metal teapot), the calculated surface area (A), the Stefan-Boltzmann constant (σ), and the temperatures (T and Tₛ) into the formula P = eσA(T⁴ - Tₛ⁴).

Step 5: Estimate the temperature drop after 30 minutes. Use the relationship Q = P × t, where Q is the heat lost, P is the power (rate of heat loss), and t is the time in seconds (30 minutes = 1800 seconds). Then, use the formula Q = mcΔT, where m is the mass of the tea (density of water is approximately 1000 kg/m³, so m = 0.55 kg), c is the specific heat capacity of water (c = 4186 J/kg·K), and ΔT is the temperature change. Solve for ΔT for each teapot.

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이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

13m

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Emissivity

Emissivity is a measure of an object's ability to emit thermal radiation compared to a perfect black body, which has an emissivity of 1. In this context, the ceramic teapot has a higher emissivity (0.70) than the shiny metal teapot (0.10), meaning it will radiate heat more effectively. This property significantly influences the rate of heat loss through radiation.

Stefan-Boltzmann Law

The Stefan-Boltzmann Law states that the power radiated by a black body is proportional to the fourth power of its absolute temperature. For real objects, this law is modified by the emissivity factor. In this problem, it helps calculate the rate of heat loss from each teapot by considering their temperatures and emissivities, allowing us to estimate how much heat they lose to the surroundings.

Heat Transfer

Heat transfer refers to the movement of thermal energy from one object to another due to a temperature difference. In this scenario, the teapots lose heat to the surrounding environment at a temperature of 20°C. Understanding the mechanisms of heat transfer, particularly radiation in this case, is essential for estimating the temperature drop of the tea over time.