Mechanical Properties of Fluids

Question

At what temperature is the root mean square speed of an atom in an argon gas cylinder equal to the rms speed of a helium gas atom at – 20 °C? (atomic mass of Ar = 39.9 u, of He = 4.0 u).

Solution

Temperature of the helium atom, T_He = –20°C= 253 K
Atomic mass of argon, M_Ar= 39.9 u
Atomic mass of helium, M_He = 4.0 u
Let, (v_rms)_Ar be the rms speed of argon.
Let (v_rms)_He be the rms speed of helium.
The rms speed of argon is given by,

open parentheses straight v subscript rms space close parentheses subscript A r end subscript space equals space square root of fraction numerator 3 R T subscript A r end subscript over denominator M subscript A r end subscript end fraction end root space space space space space space space space space space space space space space space space space space space space space space... space left parenthesis 1 right parenthesis w h e r e comma space R space equals space u n i v e r s a l space g a s space c o n s t a n t comma T subscript A R end subscript space equals space t e m p e r t a u r e space o f space a r g o n space g a s space T h e space r m s space s p e e d space o f space t h e space h e l i u m space g a s space i s space g i v e n space b y comma space open parentheses v subscript r m s end subscript close parentheses subscript H e end subscript space equals space square root of fraction numerator 3 R T subscript H e end subscript over denominator M subscript H e end subscript end fraction end root space space space space space space space space space space space space space space space space space space space space space space space space space... space left parenthesis 2 right parenthesis space G i v e n space t h a t comma space open parentheses v subscript r m s end subscript close parentheses subscript A r end subscript space equals space open parentheses v subscript r m s end subscript close parentheses subscript H e end subscript space square root of fraction numerator 3 R T subscript A r end subscript over denominator M subscript A r end subscript end fraction end root space equals space space square root of fraction numerator 3 R T subscript H e end subscript over denominator M subscript H e end subscript end fraction end root space T h e r e f o r e comma space T subscript A r end subscript over M subscript A r end subscript space equals space T subscript H e end subscript over M subscript H e end subscript T subscript A r end subscript space equals space T subscript H e end subscript over M subscript H e end subscript x space M subscript A r end subscript space space space space space space space space equals space 253 over 4 x 39.9 space space space space space space space space equals space 2523.675 space space space space space space space equals space 2.52 space x space 10 cubed space K

= 2523.675
= 2.52 × 10³ K
Therefore, the temperature of the argon atom is 2.52 × 10³K.

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Mechanical Properties Of Fluids

At what temperature is the root mean square speed of an atom in an argon gas cylinder equal to the rms speed of a helium gas atom at – 20 °C? (atomic mass of Ar = 39.9 u, of He = 4.0 u).

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