The work function for caesium atom is 1.9 eV. Calculate (a) the threshold wavelength and (b) the threshold frequency of the radiation. If the caesium element is irradiated with a wavelength 500 nm, calculate the kinetic energy and the velocity of the ejected photoelectron.
Solution and Explanation
It is given that the work function (W0) for caesium atom is 1.9 eV.
(a) From the expression,W0 = \(\frac{hc}{\lambda_0}\) , we get: λ0 = \(\frac{hc}{W_0}\)
Where, W0 = threshold wavelength
h = Planck's constant
c = velocity of radiation
Substituting the values in the given expression of W0: λ0 = \(\frac{(6.626\times10^{-34}Js)(3.0\times10^8ms^{-1})}{1.9\times1.602\times10^{-19}J}\)
λ0 = 6.53 × 10-7 m
Hence, the threshold wavelength λ0 is 653 nm.
(b) From the expression, W0 = hv0 , we get: v0 = \(\frac{W_0}{h}\)
Where, v0= threshold frequency
h = Planck's constant
Substituting the values in the given expression of W0: v0 = \(\frac{1.9\times1.602\times10^{-19}J}{6.626\times10^{-34}Js}\)
(1 eV = 1.602 × 1019 J)
W0= 4.593 × 1014 s-1
Hence, the threshold frequency of radiation (W0) is 4.593 × 10 14 s-1.
(c) According to the question: Wavelength used in irradiation = 500 nm
Kinetic energy = hc (\(\frac{1}{\lambda}-\frac{1}{\lambda_0}\)) = (6.626×10-34 Js) (3.0×108 ms-1) (λ0 - λ / λλ0) = (1.9878 × 10-26 Jm) [\(\frac{(653 - 500) 10 - 9 m}{(653) (500) 10 - 18 m2}\)]
=\(\frac{(1.9878 × 10^{-26}) (153 × 10^9)}{(653)(500)}J\) = 9.3149 × 1020 \(J\)
The kinetic energy of the ejected photoelectron = 9.3149 × 1020J
Since K.E = \(\frac{1}{2}\)mv2 = 9.3149 × 10-20 J
v = \(\frac{√2 (9.3149 × 10^{-20J})}{9.10939 × 10^{-31}}\) kg = √2.0451 × 1011 m2 s-2
v = 4.52 × 105 ms-1
Hence, the velocity of the ejected photoelectron (v) is 4.52 × 105 ms-1.
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Concepts Used:
Structure of Atom
Atomic Structure:
The atomic structure of an element refers to the constitution of its nucleus and the arrangement of the electrons around it. Primarily, the atomic structure of matter is made up of protons, electrons and neutrons.
Dalton’s Atomic Theory
Dalton proposed that every matter is composed of atoms that are indivisible and indestructible.
The following are the postulates of his theory:
- Every matter is made up of atoms.
- Atoms are indivisible.
- Specific elements have only one type of atoms in them.
- Each atom has its own constant mass that varies from element to element.
- Atoms undergo rearrangement during a chemical reaction.
- Atoms can neither be created nor be destroyed but can be transformed from one form to another.
Cons of Dalton’s Atomic Theory
- The theory was unable to explain the existence of isotopes.
- Nothing about the structure of atom was appropriately explained.
- Later, the scientists discovered particles inside the atom that proved, the atoms are divisible.
Subatomic Particles
- Protons - are positively charged subatomic particles.
- Electron - are negatively charged subatomic particles.
- Neutrons - are electrically neutral particles and carry no charge
Atomic Structure of Isotopes
Several atomic structures of an element can exist, which differ in the total number of nucleons.These variants of elements having a different nucleon number (also known as the mass number) are called isotopes of the element. Therefore, the isotopes of an element have the same number of protons but differ in the number of neutrons. For example, there exist three known naturally occurring isotopes of hydrogen, namely, protium, deuterium, and tritium.