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What is the de-Broglie wavelength of nit...

What is the de-Broglie wavelength of nitrogen molecule in air at 300K ? Assume that the molecule is moving with the root mean square speed fo molecules at this temperature. (Atomic mass of nitrogen `=14.0076U`) Plank's constant=`6.63xx10^(-34)Js` , Boltzmann constant `=1.38xx10^(-23)JK^(-1)`

Text Solution

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Here, temperature T=300K, Atomic
mass of nitrogen =14.0076U , mass of nitrogen
molecule `m=2xx14.0076=28.0152U`
`=28.0152xx1,66xx10^(-27)kg`
Let `v_(rms)` be the not mean square speed of the
nitrogen molecule at temperatureT(=300K), then
KE of nitrogen `=3/2 kT`
or `1/2mv_(rms)^2=3/2kT`
or `v_(rms)=sqrt((3kT)/m)`
De-Broglie wavelength of nitrogen molecule
`lambda=h/(mv_(rms))=h/(msqrt(3kT//m))=h/(sqrt(3mkT))`
`=(6.63xx10^(-34))/(sqrt(3xx(28.0512xx1.66xx10^(-27))xx(1.38xx10^(-23))xx300))`
`0.028xx10^-8m=0.028nm`
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What is the de-Broglie wavelength of a nitrogen molecule in air at 300 K? Assume the molecule is moving with the root-mean-square speed of molecules at this temperature. [Atomic mass of nitrogen=14.0076 u].

The temperature of an ideal gas in 3- deimensions is 300 K. The corresponding de - Broglie wavelength of the electron approximately at 300 K , is : [ m_e = mass of electron = 9 xx 10^(-31) kg h = Planck constant = 6.6 xx 10^(-34) Js k_B = Boltzmann constant = 1.38 xx 10^(-33)JK^(-1) ]

Knowledge Check

  • The temperature at which oxygen molecules have the same root mean square speed as helium atoms have at 300 K is: Atomic masses: He=4 u, 0 = 16u)

    A
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    B
    600 K
    C
    1200 K
    D
    2400 K

  • At what temperature will the rms speed of oxygen molecules becomes just sufficient for escaping from the Earth's atmosphere ?(Given : Mass of oxygen molecule (m) = 2.76 xx 10^(-26) kg. Boltzmann's constant K_B= 1.38 xx 10^(-23) JK^(-1))

    A
    ` 5.016 xx 10^(4)K`
    B
    ` 8.360 xx 10^(4)K`
    C
    ` 2.508 xx 10^(4)K`
    D
    ` 1.254 xx 10^(4)K`

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