A hydrogen-like gas is kept in a chamber having a slit of width `d = 0.01 mm`. The atom of the gas are continuously excited to a certain energy state . The excite electron make transition to lower levels , From the initial excite state to the second excited state and then from the second excited state ground state. In the process of deexcitation, photons are emitted and come out of the container through a slit. The intensity of the photons is observed on a screen placed parallel to the plane of the slit . The ratio of the angular width of the central maximum corresponding to the two transition is `25//2`. The angular width of the central maximum due to first transition is `6.4 xx 10^(-2)` radian. Find the atomic number of the gas and the principal quantum number of the inital excited state.

Answer

2

Explanation

`(1)/(lambda) = R Z ((1)/(n_(f))^(2) - (1)/(n_(1) ^(2)))`
`(lambda_(2))/(lambda_(1)) = ((1)/(9) - (1)/(n^(2)))/(8//9)`
where `n` is the principal quantum number of the initial excited state .
Angular width `theta = (2 lambda)/(d)`
`theta _(2))/(theta_(1)) = (lambda_(2))/(lambda_(1)) = (2)/(25) = (n^(2) - 9)/(8 n^(2)) implies n = 5`
`(2 lambda_(1))/(d) = theta_(1). (1)/(lambda_(1)) = (2)/(d xx theta_(1) = (2)/(6.4 xx 10^(7) m^(-1)`
`(1)/(lambda_(1)) = R Z^(2) ((1)/(9) - (1)/(n^(2)))`
`(2)/(6.4 xx 10^(7) = 1.097 xx 10^(7) z^(2) ((1)/(9) - (1)/(25))`
`z^(3) = (2)/(6.4) xx (225)/(16 xx 1.097) implies z = 2`