Will a gas sample have a root-mean-square speed of #v_(RMS) = sqrt((3RT)/m)#?

Answer 1

Note that your equation is incorrect. #R# is paired with the MOLAR mass, #M#, in #"kg/mol"#, and not the mass in #"kg"#. #k_B# is paired with the mass #m#.

Also, you cannot then invoke the ideal gas law to rewrite #upsilon_(RMS)# if it is a real gas. So this form as it is, works just fine.

#f(upsilon)dupsilon = 4pi(m/(2pik_BT))^(3//2)upsilon^2e^(-mv^2//2k_BT)dupsilon#

#color(green)(upsilon_(RMS)) = sqrt(<< upsilon^2 >>) = (int_(0)^(oo) upsilon^2f(upsilon)dupsilon)^(1//2)#

#= color(green)(sqrt((3k_BT)/m) = sqrt((3RT)/M))#

And so, you should see that #upsilon_(RMS)# comes from this distribution and must have obeyed the assumptions made in order to be valid.

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Answer 2

Wyatt Adkins

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.