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#?

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Answer 1 · 2018-06-27T19:36:04

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")#.