Can I take heat from the air and convert it to electricity? Its a summer day and the air in my house has been heated up. I could switch on my air conditioning, but then I'd be using energy from the grid in order to reduce the amount of energy in my house.
What I'd much rather do is capture the heat energy from the air, so cooling it down, and turn that heat energy into electricity. (So I could store it in a battery or sell it to the grid or use some other means of disposing of it.)
Is such an apperatus possible within our present understand of nature? Why or why not?
(I presume not, as people would be selling kits.)
Just to pre-empt a possible answer: I've seen similar conversations in the past and "inefficient" has been given as the answer as to why not. I don't understand this as an objection, because the heat energy is effectively free. The sun rudely went and gave me an excessive amount of heat energy without being asked to. Is converting heat to electricity really so ineffiecient that I'd be better off running air conditioning?
 A: I like Claudius' answer. To concisely reiterate, if you could take the heat inside your home and turn it directly into electricity, it would violate the second law of thermodynamics and be equivalent to creating perpetual motion.
Although it is nearly certain that such a device is impossible, the details of why it is impossible can be subtle. One famous attempt to do something like what you're interested in is Maxwell's Demon.
Since you're looking for an intuitive explanation of why you can't simply turn heat into electricity (without using a cold sink as people have mentioned), the best intuitive explanation I know is the "ratchet and pawl" example from Feynman's lectures. Unfortunately, I can't find Feynman's original online, but the Wikipedia article has a decent summary. (You might want to click through to read the context.)

Although at first sight the Brownian ratchet seems to extract useful
  work from Brownian motion, Feynman demonstrated that if the entire
  device is at the same temperature, the ratchet will not rotate
  continuously in one direction but will move randomly back and forth,
  and therefore will not produce any useful work. A simple way to
  visualize how the machine might fail is to remember that the pawl
  itself will undergo Brownian motion. The pawl therefore will
  intermittently fail by allowing the ratchet to slip backward or not
  allowing it to slip forward. Feynman demonstrated that if the
  temperature  of the ratchet and pawl is the same as the temperature 
  of the paddle, then the failure rate must equal the rate at which the
  ratchet ratchets forward, so that no net motion results over long
  enough periods or in an ensemble averaged sense.2 A simple but
  rigorous proof that no net motion occurs no matter what shape the
  teeth are was given by Magnasco.

A: One can operate a solar air conditioning with the electricity from the solar panels. . There are several methods in the article , some passive.
Also there are geothermal heatpumps which could be combined with solar panels to have a completely independent from the grid air conditioning/heating system. 

A: As of last week, scientists have created a thermoelectric material that increases its efficiency to 15-20%. This is still not good enough to heat or cool your house, but it gives me hope that someday in the future, we have materials so good and cheap that we can build our houses and cars with it. So we may not be able to use the heat inside the house to cool the house but we can use the temperature difference between the inside and outside to generate electricity to keep the house warm or cool. Like you, I hope that someday we will use very little energy from the grid.
As for today technology, we have windows that heat can and cool our houses if they face the sun as explained here: http://www.chelseagreen.com/content/celebrate-the-winter-solstice-by-using-windows-to-heat-your-home/
A: In order to build any thermal engine as envisioned by you, you need both a cold and a hot reservoir, such that heat can flow from the hot part to the cold part and the entropy doesn’t decrease while you’re making energy.
The efficiency of such a machine has an upper limit of $(T_{\textrm{hot}} - T_{\textrm{cold}})/T_{\textrm{cold}}$ (as given by the perfect Carnot engine).
Given that you are usually well off when you get a cold reservoir of $T_{\textrm{cold}} = 290\textrm{ K}$ on a hot summer day ($T_{\textrm{hot}} = 320\textrm{ K}$, the efficiency of your machine has an upper bound of 10%, which does not necessarily include loss due to friction, electric resistance, escaping air etc. If you include these, you get (probably) well below 1%. But, for the sake of argument, let us continue with an assumed efficiency of 10%.
What you want to know next is the maximum heat you can transfer out of your hot reservoir into your cold reservoir. For simplicity, we will assume that the cold reservoir is very large and stays at a constant temperature, heat then flows from the hot to the cold reservoir as long as $T_{\textrm{hot}} > T_{\textrm{cold}}$. The thermal energy hence available to you is $W = c_V \times \delta T \times N \times 10\%$, where $\delta T = 30 \textrm{ K}$ is the temperature difference, $N$ is the mass/number of air (particles) and $c_V$ is the heat capacity.
For air at sea-level, Wikipedia gives me $c_V \approx 29 \textrm{ J}\textrm{K}^{-1}\textrm{mol}^{-1}$ (I am using the constant-pressure one, as we cannot compress air without putting more work into it). I shall then assume that you have a really large house of $10 \times 10 \times 10 \textrm{m}^3 = 10^6 \textrm{ L } \hat{=} 4.27 \times10^4 \textrm{ mol} = N$ where the last but one equality stems from the fact that there are about $6.02 \times 10^{23}$ particles in about $23 \textrm{ L}$. Great, we can now calculate W!
$$ W = 1.03 \textrm{ kWh } \hat{=} 21 \textrm{ cent}$$
where the last equality is a top-of-my-head number I have floating around for electricity prices in Germany during the summer of 2011.
To conclude: Even assuming that you somehow manage to build a perfect Carnot engine using the heat in the air in your house and find a magical reservoir of constant temperature (some part of the earth, possibly), you would get about 20 € per year out of it (four months of high temperature and one "charge" per day).
Really, just put some solar cells on your roof and appropriate insulation on/in your walls. ☺
A: Heat is something that comes into existence ONLY when there is a temperature difference. Heat is a flowing energy. You feel heat because the difference in your body temp and the atmospheric temp. You cannot use heat between the SAME atmospheric temperatures.
A method would be to enhance the temperature using solar panels, and store the energy in batteries or in other form like use this this energy in a continuous chemical reaction and then store the "useful" gas in a pressurised cylinder.
