# The equation for the speed of sound in a gas states that v = sqrt(gammak_BT//m). Speed v is measured in "m/s", gamma is a dimensionless constant, T is temperature in kelvin ("K"), and m is mass in "kg". Find the SI unit for k_B?

Jun 27, 2018

Well, perhaps you should write out the units... this is an extremely important exercise, which you should make sure you understand...

$v = \sqrt{\frac{\gamma {k}_{B} T}{m}}$

and thus, writing out the units...

"m"/"s" = sqrt(((cdot)(?)("K"))/("kg"))

where $\left(\cdot\right)$ indicates a unitless constant.

Therefore, solving for "(?)", the units of ${k}_{B}$:

"m"^2/"s"^2 = ((?)("K"))/("kg")

$\left(\text{kg"cdot"m"^2)/"s"^2 = (?)("K}\right)$

color(blue)((?) = ("kg"cdot"m"^2)/("s"^2 cdot"K"))

Likely, this is what you are supposed to report... But as a physicist, you should recognize that

${\text{1 J" = "1 kg" cdot "m"^2"/s}}^{2}$.

Hence, we find that ${k}_{B} = 1.38065 \times {10}^{- 23}$ is alternatively in units of $\textcolor{b l u e}{\text{J/K}}$.