A spherical pot contains 0.75 L of hot coffee (essentially water)...

Question

A spherical pot contains 0.75 L of hot coffee (essentially water) at an initial temperature of 95°C. The pot has an emissivity of 0.60, and the surroundings are at 20.0°C. Calculate the coffee’s rate of heat loss by radiation.

Accepted Answer

welcome back everybody. We are taking a look at a spherical container during a room heating exercise. Now we are told that the volume of this container is two leaders were told that boiling water is being poured into it. Which temperature of the boiling water is 100 degrees Celsius, which will make the sphere also 100 degrees Celsius. Now the emissivity Of the sphere is 0.55 and we are also told that the temperature of the room or the surroundings is 14°C. And we are tasked with finding um the rate at which the water will lose heat through radiation. Now luckily we have a formula for this, right? The formula for this is going to be e times the Stefan bolts. Hman constant times the area of the cross section of the sphere, times the temperature of the sphere to the fourth minus the temperature of the surroundings to the fourth. Now, before we can even start plugging in numbers, we have to make sure we're dealing with correct constance and find the correct numbers. So, I'm gonna start out with the temperatures here. We actually were given in Celsius but we need to be working in kelvin here. So I'm gonna convert real quick. We have 100 degrees Celsius for the sphere plus 2 73. To convert. It gives us 373 kelvin. I'm gonna do the same thing with the room here, we have our given 14 degrees Celsius plus 2 73. To convert gives us kelvin right now, we need to find this area Well in order to find the area we need to know the radius but were only given the volume here. So the fact that we're working with a sphere helps. So the volume of sphere is given by four thirds times pi times are desired radius cubed. Now, before we can even start using numbers there, we have to make sure that we are dealing with the correct volume. We need to be dealing in meters cubes. Not leaders only convert this real quick. We have two leaders time one m cubed per 1000 liters gives us that the volume of the sphere is two times 10 to the negative third meters cubed. And as a reminder this is equal to four thirds pi r cubed. Now rearranging some terms here to isolate our we have that. Our radius is equal to the cube root of three times our volume of two times 10 to the negative third. All over over four pi. Which when you plug this into your calculator, you get that our radius is 0. m. Now that we have that we can find the area of the cross section of the sphere which is just a circle. So it's going to be area is pi r squared which is equal to pi times 0.7816 squared. Which when you plug into your calculator, we get that area is 0.7677 m squared. Great. Now we have everything in the right units and in the right term. So let's go ahead and find the radiation here. We have um, our emissivity of 0.55 times the Stefan bolts man constant of 5.67 times 10 to the negative eight, uh, times our area of 0. Times the temperature of our sphere, which is 373 Kelvin to the power of -287 to the power of four is equal to 30.1 watts. When you plug it into your calculator corresponding to our answer choice of E. Thank you all so much for watching. Hope this video helped. We will see you all in the next video.

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