If the absolute temperature of a gas is tripled, what happens to the root-mean-square speed of the molecules?

1 Answer
Jun 20, 2017

Answer:

It increases by a factor of #sqrt3#

Explanation:

The root-mean-square speed #u_"rms"# of gas particles is given by the equation

#u_"rms" = sqrt((3RT)/(MM))#

where

  • #R# is the universal gas constant, for this case #8.314("kg"·"m"^2)/("s"^2·"mol"·"K")#

  • #T# is the absolute temperature of the system, in #"K"#

  • #MM# is the molar mass of the gas, in #"kg"/"mol"#

The question is nonspecific for which gas, but we're just asked to find what generally happens to the r.m.s. speed if only the temperature changes, so we'll call the quantity #(3R)/(MM)# a constant, #k#:

#u_"rms-1" = sqrt(kT)#

If the temperature is tripled, then this becomes

#u_"rms-2" = sqrt(3kT)#

To find what happens, let's divide this value by the original equation:

#(u_"rms-2")/(u_"rms-1") = (sqrt(3kt))/(sqrt(kt)) = color(red)(sqrt3#

Thus, if the temperature is tripled, the root-mean-square speed of the gas particles increases by a factor of #color(red)(sqrt3#.