Question

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`?

Answer 1

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

---

You had...

`v = sqrt((gammak_BT)/m)`

and thus, writing out the units...

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

where `(cdot)` indicates a unitless constant.

Therefore, solving for "`(?)`", the units of `k_B`:

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

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

`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

`"1 J" = "1 kg" cdot "m"^2"/s"^2`.

Hence, we find that `k_B = 1.38065 xx 10^(-23)` is alternatively in units of `color(blue)("J/K")`.