Answer to Question #154499 in Physics for Abhinaba Banerjee

Question #154499

Calculate the root mean square velocity of air molecules at N.T.P.(Density of air = 1.293 Kg/m³).


1
Expert's answer
2021-01-10T18:27:35-0500

The root mean square velocity of air molecules can be found from the formula:


vrms=3RTM.v_{rms}=\sqrt{\dfrac{3RT}{M}}.

Let's express RTM\dfrac{RT}{M} in terms of PP and ρ\rho. From the ideal gas law, we have:


PV=nRT,PV=nRT,

here, n=mM.n = \dfrac{m}{M}.

Let's divide both sides of the equation by VV:


P=mVRTM.P=\dfrac{m}{V}\dfrac{RT}{M}.

Since ρ=mV\rho=\dfrac{m}{V} we have:


P=ρRTMP=\rho\dfrac{RT}{M}

Finally, we get:


RTM=Pρ.\dfrac{RT}{M}=\dfrac{P}{\rho}.


Then, we can calculate the root mean square velocity of air molecules at normal temperature and pressure:


vrms=3RTM=3Pρ,v_{rms}=\sqrt{\dfrac{3RT}{M}}=\sqrt{\dfrac{3P}{\rho}},vrms=31.013105 Pa1.293 kgm3=485 ms.v_{rms}=\sqrt{\dfrac{3\cdot 1.013\cdot10^5\ Pa}{1.293\ \dfrac{kg}{m^3}}}=485\ \dfrac{m}{s}.

Answer:

vrms=485 ms.v_{rms}=485\ \dfrac{m}{s}.


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