Question

the temperature at which
the RMS velocity of helium is
equal to the RMS velocity of
methane at 127°C​

Answer 1

Solution :

To Find :-

The temperature at which the RMS velocity of helium is equal to the RMS velocity of methane at 127°C

Formula :-

Formula of RMS velocity in terms of temperature of gas is given by

$\underline{\underline{\boxed{\bf{\purple{V_{RMS}=\sqrt{\dfrac{3RT}{M_w}}}}}}}$

• R denotes gas constant
• T denotes temperature of gas (in K)
• Mw denotes molecular weight of gas

▪ Mw of Helium(He) = 4g/mol

▪ Mw of methane(CH$_4$) = 16g/mol

Conversation :-

⏭ K = °C + 273

⏭ Temp. of methane = 127 + 273 = 400K

Calculation :-

$\Rightarrow\bf\:(V_{RMS})_{He}=(V_{RMS})_{CH_4}\\ \\ \Rightarrow\sf\:\sqrt{\dfrac{3RT_{He}}{(Mw)_{He}}}=\sqrt{\dfrac{3RT_{CH_4}}{(M_w)_{CH_4}}}\\ \\ \Rightarrow\sf\:\dfrac{T_{He}}{(M_w)_{He}}=\dfrac{T_{CH_4}}{(M_w)_{CH_4}}\\ \\ \Rightarrow\sf\:\dfrac{T_{He}}{4}=\dfrac{400}{16}\\ \\ \Rightarrow\sf\:T_{He}=\dfrac{400\times 4}{16}\\ \\ \Rightarrow\sf\:T_{He}=\dfrac{1600}{16}\\ \\ \Rightarrow\underline{\underline{\boxed{\bf{\pink{T_{He}=100K=-173\degree{C}}}}}}$