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Most people find waterbeds uncomfortable unless the water the waterature is maintained at about $85^{\circ} \mathrm{F}$ . Unless it is heated, a waterbed that contains 892 $\mathrm{L}$ of cools from $85^{\circ} \mathrm{F}$ to $72^{\circ} \mathrm{F}$ in 24 hours. Estimate the amount of electrical energy required over 24 hours, in kWh, to keep the bed from cooling. Note that 1 kilowatt-hour (kWh) =$3.6 \times 10^{6} \mathrm{J},$ and assume that the density of water is 1.0 $\mathrm{g} / \mathrm{mL}$ (independent of temperature). What other assumptions did you make? How did they affect your calculated result (i.e., were they likely to yield positive or negative errors)?

$E(\mathrm{kWh})=7.464 \mathrm{kWh}$

Chemistry 101

Chapter 5

Thermochemistry

Nicholas A.

November 9, 2021

Most people find waterbeds uncomfortable unless the water temperature is maintained at about 85 °F. Unless it is heated, a waterbed that contains 892 L of water cools from 85 °F to 72 °F in 24 hours. Estimate the amount of electrical energy required over

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Brown University

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Hi there. In this problem, we are trying to calculate how much energy it would take to prevent this waterbed from cooling down. That means I want to use the equation that allows us to calculate energy when we know the specific heat of the substance. In this case, it is water multiplied by the mass of the water times the change in temperature. Using this equation will allow me to calculate how much energy is required to keep this bed from cooling down overnight. Um, a couple of things, first of all, is our final temperature. The temperature we would like to attain is 85 F. Our values of specific heat are in Celsius. So the first thing I want to do is convert this temperature toe a Celsius temperature. I'm going to use the equation where we take 5/9 the Fahrenheit temperature minus 32. Okay. And that gives me 29.4 degrees Celsius. So 85 F is 29 0.4 degrees Celsius, doing the same thing with the initial temperature. It's initially starting out at that colder temperature or cooler temperature, I guess I would say, and that is 72 F. Using my same equation 5/9 times 70 to minus 32 I get a Celsius temperature of 22.2 degrees Celsius. Now I can go ahead and use my equation. I am trying to calculate the amount of energy to keep this waterbed At 85 F. It is water filling up this waterbed. So I am going to use the specific heat of liquid water, which we confined in the table. It's 4.184 Jules over Graham's degree Celsius. There's that Celsius temperature, which is why I had to convert my temperature values to Celsius. Next for the water. I have 892 leaders, but I need to know how many grams of water I have, and we are given the density in grams per milliliter. But I need to get to milliliters, So I am going to do a quick conversion here. For every one leader of water, there is 1000 male leaders, and since this is water, we were told to assume that the density of the water remained constant at 1 g per mil. Later, doing this, math leaders would cancel mill leaders would cancel and this gives us the grams of water. Finally, we need to calculate by delta t the change in Celsius temperature. We're going to take the final temperature that we're trying to attain, which is 29.4 degrees Celsius ever going to subtract the initial temperature 22.2 degrees Celsius. Alright. Calculating this answer then gives me 2.69 times 10 to the seventh jewels of energy. This problem wants us to present the answer and kilowatt hours and they give us a conversion. They tell us that one kilowatt hour is equal to 3.6 times 10 to the six jewels calculating our final answer, then gives May 7.46 kilowatt hours are required are required to keep this bed at the temperature the 85 degrees Celsius that we wanted. Right as we think about this, there might be some error in this answer because not all of the energy is going to be used to heat the water. Some of that energy is going to be going into heating the bed itself, the frames, things like that. So this answer is probably a bit lower than what it would actually be correct because there's gonna be more energy needed to heat up some of the other things that we did not calculate again. Like the bed frame, the bed itself, any of the other things that are touching that water. All right, well, hey, that would be your answer. Your answers for this. Thanks so much for watching.

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