Why does most of the heat transferred on Earth come from the infrared part of the electromagnetic spectrum? Why does the most of the heat transferred on Earth come from infrared part of electromagnetic spectrum?
 A: The expectation value of frequency $\nu$ in Planck's law can be obtained via
$$\langle\nu\rangle=\frac{\int_0^\infty2h\nu^4c^{-2}\left(\exp\left(\frac{h\nu}{kT}\right)-1\right)^{-1}\,d\nu}{\int_0^\infty2h\nu^3c^{-2}\left(\exp\left(\frac{h\nu}{kT}\right)-1\right)^{-1}\,d\nu}=\frac{360 k T \zeta (5)}{\pi ^4 h}$$
where $\zeta$ is the Riemann Zeta function.
For $T=273\text{K}$, this becomes $21.7993\text{THz}$, or $13.75\mu\text{m}$, well in the infrared range.
A: Heat transferred on Earth is a little vague...let's break it down


*

*Heat transferred to / from you.  Your heat transfer to the environment is a combination of radiation, convection, and conduction.  Conduction to air is pretty small, so actually it is closer to 30% radiation and 70% convection.  A good number to use for free convection (no wind) is 4 W/sq-m K (Heat Transfer, Holman - all time favorite heat transfer book given to me by my brother-in-law).  This is why metal feels cold.  You are used to interfacing to "room temperature" air.  When you touch metal, its high thermal conductivity, sucks the heat energy out of you and thus feels cold even though it is room temperature.

*Heat transfer to / from Earth from the Universe.  To understand this, it is useful to understand what is making the heat energy, what is absorbing the heat energy, and what is losing the heat energy.  Energy coming into the Earth is dominated by blackbody radiation from the sun.  Take a look at the blackbody curve for a perfect blackbody at the temperature of the sun:


http://apollo.lsc.vsc.edu/classes/met130/notes/chapter2/graphics/plank_sun_closer_look.jpg
This tells you that the peak of the blackbody curve is in the visible spectrum (thank goodness our eyes are sensitive there, right... otherwise it'd be awfully dark).  However, there is plenty of energy coming at the Earth that is not visible (integrate all the energy over the curve).
So, we've established that energy is coming into the Earth from visible, IR, radio, UV, cosmic rays, ... but most of it is actually near the visible spectrum.
The Earth's core is still an energy source (residual heat from the formation of the planet from kinetically high energy particles... just simply stuff moving fast = hot).  Let's neglect that for now.
Finally, the Earth gives off energy to the Universe.  At what wavelength is this?  Well, as  wrote, Earth is pretty close to 300K.  Check out this blackbody comparison:
http://apollo.lsc.vsc.edu/classes/met130/notes/chapter2/graphics/earth_sun_plank.jpg
From here, you can see that the wavelengths that have the most emission from the Earth to the Universe are 8 - 12 um or what some call the LWIR (Long Wave infrared).
Finally, we come to the "alpha over e" expression.  Energy comes in at visible wavelengths, but emits at the LWIR (alpha refers to absorption while e refers to emissivity).  So heat transfer between the Earth and the Universe is dominated by how much visible light the Earth absorbs and how much LWIR it emits.  So, if you want to heat up the Earth, paint it black.  If you want to cool it down, paint it with some material that emits in the LWIR (this is not always "black" to you and I in the visible spectrum).
Want to see what the Earth looks like at these LWIR wavelengths?  Check out the NASA MODIS website.
If you really want to think about this at the Quantum Mechanical level, I suggest Thermal Physics by Kittel and Kroemer.  It will give you a simple outline of how to think about Blackbody radiation from a Quantum Mechanical level.  They basic argument is that we are all coupled to each other (and everything).  Blackbody radiation represents that quantum mechanical coupling to all matter.  This continuum of states is expressed as the "wavelengths" of coupled energy.
By the way, this is what makes Dark Matter so interesting.  How is it that there is mass in the Universe that is not coupled quantum mechanically through blackbody radiation?
