How to express $k_BT$ as a length scale? I am wondering how we can use $k_BT$ to define a length scale?
I mean it is a unit of energy, not of length. So how would that transformation work?
 A: $\frac{hc}{k_BT}$ is a length, where $h$ is Planck’s constant and $c$ is the speed of light.
Apart from a numerical factor of order 1, this is the thermal de Broglie wavelength for a photon gas at temperature $T$, the critical wavelength at which quantum effects begin to dominate.
Also, apart from another numerical factor of order 1, this is the Schwarzschild radius of a black hole with Hawking temperature $T$.
A: You don't give the context, but this idea tends to be used in scattering experiments where the particles are either massless (e.g. photons) or highly relativistic so their rest mass is negligible compared to their energy. In that case the de Broglie wavelength is related to the energy by:
$$ \lambda = \frac{hc}{E} \tag{1} $$
And as you say, $kT$ has the units of energy so you'd substitute it for the energy.
Your use of $kT$ suggests thermodynamics rather than particle scattering, in which case a plausible relationship might be Wien's law for the peak black body wavelength:
$$ \lambda = \frac{b}{T} \tag{2} $$
The two equations give wavelengths of about the same order of magnitude. Wien's constant $b$ is about a factor of $6$ smaller than $hc/k$.
A: It's not really clear what you're asking, since you can no more "convert" energy into length as you could convert mass to length -- they're fundamentally distinct. However, I'll assume that what you want is something like this: As you note, $k_BT$ has dimensions of energy, equivalent to $M L^2 / T^2$, where $M$, $L$, and $T$ are mass, length, and time, respectively. If you wanted to somehow convert $k_B T$ into a length, you'd have to divide this energy by some quantity with dimensions of $M L / T^2$.
