An electron from various excited states of hydrogen atom emit radiation to come to the ground state. Let `lambda_n, lambda_g` be the de Broglie wavelength of the electron in the n-th state and the ground state respectively. Let `Lambda_n` be the wavelength of the emitted photon in the transition from the n-th state to the ground state. For large n,(A,B are constants)

Options

A. `Lambda_n^2~~A+Blambda_n^2`
B. `Lambda_n^2~~lambda`
C. `Lambda_n~~A+B/lambda_n^2`
D. `Lambda_n~~A+Blambda_n`

Answer

C

Explanation

we know,`K_n=(h^2)/(2mlambda_n^2) and K_g=(h^2)/(2mlambda_g^2)`
`therefore E_n-E_g=K_g-K_n[because E_n=-K_n,E_g=-K_g]`
`or, (hc)/(Lambda_n)=(h^2)/(2m)(1/(lambda_g^2)-1/(lambda_n^2))`
`or, Lambda_n=(2mc)/h((lambda_g^2lambda_n^2)/(lambda_n^2-lambda_g^2))=(2mclambda_g^2)/(h)(1-(lambda_g^2)/(lambda_n^2))^(-1)`
`conv(2mclambda_g^2)/(h)(1+(lambda_g^2)/(lambda_n^2)) conv((2mclambda_g^2)/(h))+((2mclambda_g^4)/(h))1/(lambda_n^2)`
=`A+B/(lambda_n^2)`
where,A=`(2mclambda_g^2)/(h),B=(2mclambda_g^2)/h`