By Kinetic Theory Of Gases, Pressure Times Volume Equals (2/3)(Total Number Of Particles Present) Times (0.5m×(v2)bar) Which In Turn Equals (Total Number Of Particles Present) Times Boltzmann's Constant Times Kelvin Temperature. That is, PV = (2/3)N × (0.5m(v2)bar) = NkT.
Reduce (2/3)N × (0.5m(v2)bar) = NkT to T = (2/3k)×(0.5m(v2)bar).
Atomic Mass or M for Neon equals 20.18E-3 Kilograms Per Mole.
Avogadro's Number NA equals 6.022136736E23 Particles Per Mole.
Boltzmann's Constant k equals 1.38065812E-23 Joules Per Kelvin.
Also, m equals Atomic Mass/Avogadro's Number, equal here to 20.18E-3 Kilograms Per Mole/6.022136736E23 Atoms Per Mole or 3.350970077E-26 Kilograms Per Atom.
The Average Translational Kinetic Energy sought is equal to (0.5m(v2)bar).
For T = 354 Kelvin, write 354 = (2/3/1.38065812E-23)(0.5m×(v2)bar) which will equate
(0.5m(v2)bar) to 7.331294617E-21 Joules.
Root-Mean-Square Speed or vrms is obtained from T = (2/3k)×(0.5m(v2)bar), rewritten as
(3kT/m) = (v2)bar. Then √(v2)bar = √[(3×1.38065812E-23×354) ÷ (20.18E-3/6.022136736E23)] which reduces to 661.4850894 Meters Per Second.
To triple the speed of the particles or atoms, equate 3(661.4850894) to √[(3kT)/(M/NA)] or 3(661.4850894) =
√[(3×1.38065812E-23×T) ÷ (20.18E-3/6.022136736E23)] which gives the value of T sought as 3186 Kelvin.