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Same gas is filled in two containers of same volume, same temperature and with pressure of ratio 1 : 2. The ratio of their rms speeds is:
1. 1 : 2
2. 2 : 1
3. 1 : 4
4. 1 : 1

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Correct Answer - Option 4 : 1 : 1

CONCEPT:

Root mean square velocity of the gas

  • Root mean square velocity (RMS value)is the square root of the mean of squares of the velocity of individual gas molecules.
  • It is given as,

\(⇒ v_{rms}=\sqrt{\frac{3RT}{M}}\)

Where vrms= Root-mean-square velocity, M = Molar mass of the gas (Kg/mole), R= Molar gas constant and T = Temperature in Kelvin

CALCULATION:

  • Since the same gas is filled in two containers of the same volume, same temperature, and with the pressure of ratio 1 : 2.

Given M1 = M2 = M, V1 = V2 = V, T1 = T2 = T and \(\frac{P_{1}}{P_{2}}=\frac{1}{2}\)

  • The rms speed of the gas is given as,

\(⇒ v_{rms}=\sqrt{\frac{3RT}{M}}\)     -----(1)

  • The rms speed for container 1 is given as,

\(⇒ v_{rms1}=\sqrt{\frac{3RT}{M}}\)     -----(2)

  • The rms speed for container 2 is given as,

\(\Rightarrow v_{rms2}=\sqrt{\frac{3RT}{M}}\)     -----(3)

By equation 2 and equation 3,

\(\Rightarrow\frac{v_{rms1}}{v_{rms2}}=\frac{1}{1}\)

Hence, option 4 is correct.

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