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(a) On a winter day, a certain house loses $5.00 \times 10^8 \textrm{ J}$ of heat to the outside (about 500,000 Btu). What is the total change in entropy due to this heat transfer alone, assuming an average indoor temperature of $21.0^\circ\textrm{C}$ and an average outdoor temperature of $5.00^\circ\textrm{C}$? (b) This large change in entropy implies a large amount of energy has become unavailable to do work. Where do we find more energy when such energy is lost to us?
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Final Answer
  1. $9.78 \times 10^4 \textrm{ J/K}$
  2. Indirectly, energy comes from the sun, which is stored as fossil fuels that are burned, or which makes electrical energy via hydro, photovoltaic, etc.
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OpenStax College Physics Solution, Chapter 15, Problem 47 (Problems & Exercises) (2:53)

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<i>This is College Physics Answers with Shaun Dychko.</i> The total change in entropy and I have <i>T</i> here for total is going to be the change of entropy of the inside of the house, plus the change of entropy of the outside of the house. Now in general, change of entropy is the amount of heat transferred into the system, divided by the absolute temperature of the system. And so, we can substitute for <i>Delta Si</i> and <i>Delta So</i> by replacing them with heat transferred divided by the temperature inside, plus heat transferred divided by the temperature outside. Now inside the house, there is a loss of heat. And so we have this minus sign in front of this <i>Q</i> that we're given. And <i>Q</i> is five times 10 to the eight Joules. And then we add to that a plus, a positive <i>Q</i> because the outside is having the heat transferred into it. So we have <i>Q over To</i>. And then we'll factor the <i>Q</i> out and now let's just change the order of the terms as well. So we have a positive minus a negative. And so, well it's not really "minus a negative" that's a bit of a bad way to say it. So it's a positive and then a minus. So we have <i>Q</i>, times one over temperature of the outside, minus one over temperature of the inside. So that's five times 10 to the eight Joules transferred which we'll multiply by one over five degrees outside, converted into Kelvin by adding 273.15, minus one over 21 degrees Celsius inside, also converted into Kelvin. And this makes 9.78 times 10 to the four Joules per Kelvin, is the total change in entropy of the system resulting from this transfer of heat from the inside of the house to the outside. So, now the question is, so a lot of energy has become unavailable to do work, and where do we find more energy? Well, we can burn more gasoline, or not gasoline in the house. Typically sometimes you burn diesel fuels. Some houses are heated with oil so to speak. They have an oil drum outside then it will still live inside. But typically, natural gas is what's burned, or what we could use some electrical energy from hydro or photovoltaic. But anyway, indirectly, one way or the other, energy always comes from the sun. The only exception to that is geothermal energy. But if you're burning fossil fuels, you're burning carbon that was originally grown using energy from the sun. And fossil fuels are basically a storage, a medium, for the sun's energy. And so, there we go.