The `k_alpha` X-rays of aluminium (Z = 13 ) and zinc ( Z = 30) have wavelengths 887 pm and 146 pm respectively. Use Moseley's law `sqrt v = a(Z - b) to find the wavelength of the `K_alpha` X-ray of iron (Z = 26).

Explanation

Given ` lambda=887p m `
` f=c/lambda=(3xx10^8)/(887xx10^-12) `
` 3.382xx10^17 `
` 3.382xx10^16 `
` lambda_2=146p m `
` f=(3xx10^8)/(146xx10^-12) `
` 0.02054xx10^20 `
` =2.054xx10^18 `
We know `f=a(Z-b) `
` 5.815xx10^8=a(13-b).....(1) `
` 21.4331xx10^9=a(30-b) .......(2) `
Dividing(1) by (2)
` (13-b)/(30-b)=(5.815xx10^-1)/21.4331 =0.4057 `
` rArr 30xx0.4057-0.4057b=13-b `
` rArr 12.171-0.4057b+b=13 `
` b=0.829/0.5943=1.39491 `
` a=(5.815xx10^8)/11.33 `
` =0.51323xx10^8=5xx10^7 `
For 'Fe'
` f=5xx10^7(26-1.39) `
` =5xx24.61xx10^7 `
` =123.05xx10^7 `
` and c/lambda=5141.3xx10^14 `
` lambda=(3xx10^8)/(5141.3xx10^14) `
` 0.000198xx10^-5m `
` =198xx10^-12=198p m`