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Calculate Wavelength of Radiation in Photoelectric Effect

The surface of sodium metal is irradiated with radiation of wavelength x nmx\text{ nm}. The kinetic energy of ejected electrons is 2.8×10−20 J2.8 \times 10^{-20}\text{ J}. The work function of sodium is 2.3 eV2.3\text{ eV}. The value of xx is ‾×102 nm\underline{\quad\quad} \times 10^2\text{ nm}. (Nearest integer) (Given : h=6.6×10−34 J sh = 6.6 \times 10^{-34}\text{ J s} ; 1 eV=1.6×10−19 J1\text{ eV} = 1.6 \times 10^{-19}\text{ J} ; c=3.0×108 m s−1c = 3.0 \times 10^8\text{ m s}^{-1})

Official Numerical Answer5

Topics & Concepts

Structure of AtomBohr Model

Step-by-Step Solution

According to Einstein's photoelectric equation, the energy of an incident photon (EE) is equal to the sum of the work function (Φ\Phi) of the metal and the maximum kinetic energy (KEKE) of the ejected electrons:

E=Φ+KEE = \Phi + KE

Step 1: Convert the work function to Joules

Given the work function Φ=2.3 eV\Phi = 2.3\text{ eV} and 1 eV=1.6×10−19 J1\text{ eV} = 1.6 \times 10^{-19}\text{ J}:

Φ=2.3×1.6×10−19 J=3.68×10−19 J\Phi = 2.3 \times 1.6 \times 10^{-19}\text{ J} = 3.68 \times 10^{-19}\text{ J}

Step 2: Calculate the total energy of the incident radiation

Given the kinetic energy of the ejected electrons KE=2.8×10−20 J=0.28×10−19 JKE = 2.8 \times 10^{-20}\text{ J} = 0.28 \times 10^{-19}\text{ J}:

E=3.68×10−19 J+0.28×10−19 J=3.96×10−19 JE = 3.68 \times 10^{-19}\text{ J} + 0.28 \times 10^{-19}\text{ J} = 3.96 \times 10^{-19}\text{ J}

Step 3: Calculate the wavelength (λ\lambda) of the incident radiation

The energy of a photon is related to its wavelength by:

E=hcλ  ⟹  λ=hcEE = \frac{hc}{\lambda} \implies \lambda = \frac{hc}{E}

Substituting the given values (h=6.6×10−34 J sh = 6.6 \times 10^{-34}\text{ J s} and c=3.0×108 m s−1c = 3.0 \times 10^8\text{ m s}^{-1}):

λ=6.6×10−34×3.0×1083.96×10−19\lambda = \frac{6.6 \times 10^{-34} \times 3.0 \times 10^8}{3.96 \times 10^{-19}}

λ=19.8×10−263.96×10−19=5.0×10−7 m\lambda = \frac{19.8 \times 10^{-26}}{3.96 \times 10^{-19}} = 5.0 \times 10^{-7}\text{ m}

Step 4: Convert the wavelength into nanometers (nm\text{nm})

Since 1 m=109 nm1\text{ m} = 10^9\text{ nm}:

λ=5.0×10−7×109 nm=500 nm\lambda = 5.0 \times 10^{-7} \times 10^9\text{ nm} = 500\text{ nm}

Thus, the wavelength of radiation is x nm=500 nmx\text{ nm} = 500\text{ nm}, which gives x=500x = 500.

Expressing xx in the required format: x=5×102x = 5 \times 10^2

Hence, the value to fill in the blank is 5.

Calculate Wavelength of Radiation in Photoelectric Effect | Chemistry PYQ Solution - JEE Challenger