The energy of a single photon of a particular radiation is 3.96 10⁻¹⁹ ~J . If the wavenumber of this radiation is expressed as x 10³ ~cm ⁻¹ , the value of x is ________. (Given: h = 6.6 10⁻³⁴ ~J~s , c = 3 10⁸ ~m~s ⁻¹ )

Step-by-step solution

According to Planck's quantum theory, the energy of a photon is related to its wavenumber by: E = h c Rearranging for wavenumber ( ): = E h c Substituting the given values: = 3.96 10⁻¹⁹ 6.6 10⁻³⁴ 3 10⁸ = 3.96 10⁻¹⁹ 19.8 10⁻²⁶ ~m ⁻¹ = 0.2 10⁷ ~m ⁻¹ = 2 10⁶ ~m ⁻¹ Since 1 ~m = 100 ~cm , the unit conversion for reciprocal length is 1 ~m ⁻¹ = 10⁻² ~cm ⁻¹ . = 2 10⁶ 10⁻² ~cm ⁻¹ = 2 10⁴ ~cm ⁻¹ Expressing this in the required format: = 20 10³ ~cm ⁻¹ Thus, the value of x is 20 . Answer: 20