An ideal spherical blackbody 24.0 cm in diameter is maintained at 225^∘ C by an internal electri

An ideal spherical blackbody 24.0 cm in diameter is...

An ideal spherical blackbody 24.0 $\mathrm{cm}$ in diameter is maintained at $225^{\circ} \mathrm{C}$ by an internal electrical heater and is immersed in a very large open-faced tank of water that is kept boiling by the energy radiated by the sphere. You can neglect any heat transferred by conduction and convection. Consult Table 17.4 as needed. (a) At what rate, in $\mathrm{g} / \mathrm{s},$ is water evaporating from the tank? (b) If a physics-wise thermophile organism living in the hot water is observing this process, what will it measure for the peak intensity (i) wavelength and (ii) frequency of the electromagnetic waves emitted by the sphere?

Key Concepts

Stefan?Boltzmann Law

The Stefan?Boltzmann Law relates the power radiated by a blackbody to its temperature and surface area. Specifically, it states that the total energy radiated per unit time is proportional to the fourth power of the absolute temperature. This concept is essential for understanding how much energy a heated object emits as electromagnetic radiation and is widely used in calculating radiation heat transfer in thermal systems.

Latent Heat of Vaporization

Latent heat of vaporization is the amount of energy required to convert a unit mass of liquid into vapor without changing its temperature. It is a key concept in thermodynamics and heat transfer, especially when evaluating phase changes, such as evaporation. In problems involving energy balance, the latent heat allows one to relate the energy input or output to the corresponding change in mass due to phase change.

Wien’s Displacement Law

Wien’s Displacement Law describes the relationship between the temperature of a blackbody and the wavelength at which it emits radiation most intensely. Specifically, it states that the peak wavelength is inversely proportional to the absolute temperature. This law is critical for predicting the spectral characteristics of thermal radiation in various systems, informing applications ranging from astrophysics to thermal imaging.

Relationship between Wavelength and Frequency

This concept arises from the fundamental relationship in wave physics that the frequency of a wave is inversely proportional to its wavelength, given a constant speed of propagation (such as the speed of light in a vacuum). It is crucial for converting the peak wavelength from Wien’s Law to its corresponding peak frequency, thereby connecting the two perspectives of electromagnetic radiation spectra.

Transcript

00:01 In this particular question, we are assuming we have a very large tank and we have a black body object that is submerged inside of a pool of tank of water.

00:21 We want to find how much water is being evaporated due to only the black body radiation.

00:33 That is actually in emitting from this body.

00:38 Now we're going to ignore everything that is actually transferred by conduction or convection.

00:46 So only radiation transfer of energy.

00:51 So kind of like they are not really touching, right, except that they are, but we ignore the conduction part.

01:04 So what is the amount of energy being transferred per second? we can use the stefan bozeman's law for the power radiated.

01:19 There is the power that's the power is given by the stiffel bozeman's constant, multiplied by the surface area, what about by temperature to power 4.

01:33 Now the thing is, the water itself gains this amount of energy from the sphere but it also loses a certain amount of energy based on its own temperature right so this is the water's area surface area and water's temperature now we're going to assume that it only radiates its power in this area right it radiates back to the sphere so they will have the same surface area, a, although it doesn't really make sense, right, but we don't know what is this surface area of this.

02:24 Now we're going to assume maybe it's perfectly insulated, but this is not given.

02:31 So we cannot use that surface area, we only can use the surface area of the sphere.

02:37 Right, and of course the temperature of the water.

02:45 So now we can equate this, right, this would be the power of the water, right, so the energy being absorbed by the water must be equals to the rates of change of our mass, multiplied by the latent heat of vaporization.

03:24 Right since this is what causes the water to vaporize alright and if we multiply the mass change in mass times the vaporization this will give us the energy right total energy absorbed and over time will give us power so actually we can equate these two together.

04:08 So we have sigma a times temperature of the sphere to power minus temperature of the water.

04:20 We are assuming that the water is at boiling point already, which is basically 100 degrees celsius.

04:45 So we can rearrange this equation to find the rate of change of our mass, or water just have to divide the latent heat of vaporization over and we can replace the surface area a for pi r square where r would be half of the diameter that's given so that's 0 .12 meters sigma the constant 5 .6 7 temperature of the sphere is 225 degrees.

05:46 Just 498 kelvin's.

05:47 We take 273 plus 225...

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