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

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

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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...

$v = \sqrt{\frac{γ k_{B} T}{m}}$ $v = sqrt((gammak_BT)/m)$

and thus, writing out the units...

$\frac{m}{s} = \sqrt{\frac{(\cdot) (?) (K)}{kg}}$ $"m"/"s" = sqrt(((cdot)(?)("K"))/("kg"))$

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

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

$\frac{m^{2}}{s^{2}} = \frac{(?) (K)}{kg}$ $"m"^2/"s"^2 = ((?)("K"))/("kg")$

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

$(?) = \frac{kg \cdot m^{2}}{s^{2} \cdot K}$ $color(blue)((?) = ("kg"cdot"m"^2)/("s"^2 cdot"K"))$

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

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

Hence, we find that $k_{B} = 1.38065 \times 10^{- 23}$ $k_B = 1.38065 xx 10^(-23)$ is alternatively in units of $J/K$ $color(blue)("J/K")$ .

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The equation for the speed of sound in a gas states that v = sqrt(gammak_BT//m)v=√γkBT/mv = sqrt(gammak_BT//m). Speed vvv is measured in "m/s"m/s"m/s", gammaγgamma is a dimensionless constant, TTT is temperature in kelvin ("K"K"K"), and mmm is mass in "kg"kg"kg". Find the SI unit for k_BkBk_B?

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