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The bohrradius is given by a(0) = (epsil...

The bohrradius is given by `a_(0) = (epsilon_(0)h^(2))/(pi m e^(2))` verify that the KHS has dimesions of length

Text Solution

Verified by Experts

The correct Answer is:
A
`a= (in_0h^2)/(pime^2) = (A^2T^2(ML^2T^-1))/(M^2L^3T^-2) = L`
`=(M^2L^2T^-2)/(M^2L^3T^-2) = L`
Clearly `a_0` has dimensions of length.
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The dimensions of (1)/(epsilon_(0))(e^(2))/(hc) are

Hydrogen Atom and Hydrogen Molecule The observed wavelengths in the line spectrum of hydrogen atom were first expressed in terms of a series by johann Jakob Balmer, a Swiss teacher. Balmer's empirical empirical formula is 1/lambda=R_(H)(1/2^(2)-1/n^(2))," " N=3, 4, 5 R_(H)=(Me e^(4))/(8 epsilon_(0)^(2)h^(3) c)=109. 678 cm^(-1) Here, R_(H) is the Rydberg Constant, m_(e) is the mass of electron. Niels Bohr derived this expression theoretically in 1913. The formula is easily generalized to any one electron atom//ion. Determine the loest energy and the radius of the Bohr orbit of the muonic hydrogen atom. Ignore the motion of the nucleus in your calculation. The radius of the Bohr orbit of a hydrogen atom ("called the Bohr radius", a_(0)=(epsilon_(0)h^(2))/(m_(e)e^(2)prod) "is" 0.53 Å) The classical picture of an ''orbit'' in Bohr's theory has now been replaced by the quantum mechanical nation of an 'orbital'. The orbital psi 1 sigma 1s (r) for the ground state of a hydrogen atom is given by psi 1 s (r)=1/sqrt(proda_(0)^(3)) e^(r/a_(0)) where r is the distance of the electron from the nucleus and a_(0) is the Bohr radius.

Knowledge Check

  • The dimesions of (mu_(0) epsilon_(0))^(-1//2) are

    A
    `[L^(-1)T]`
    B
    `[LT^(-1)]`
    C
    `[L^(-1//2)T^(1//2)]`
    D
    `[L^(1//2)T^(-1//2)]`

  • The dimensions of (mu_(0)epsilon_(0))^(-1//2) are

    A
    `[L^(1//2) T^(-1//2)]`
    B
    `[L^(-1)T]`
    C
    `[LT^(-1)]`
    D
    `[L^(1//2)T^(1//2)]`

  • A quantity X is given by (me^(4))/(8epsilon_(0)^(2)ch^(3)) where m is the mass of electron, e is the charge of electron, epsilon_(0) is the permittivity of free space, c is the velocity of light and h is the Planck's constant. The dimensional formula for X is the same as that of

    A
    length
    B
    frequency
    C
    velocity
    D
    wave number

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