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What is meant by hybridisation of atomic orbitals? Describe the shapes of $ sp,\;s{p^2},\;s{p^3}\; $ hybrid orbitals:

Answer
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Hint: Hybridization is the mixing of atomic orbitals to form new types of orbitals. The new orbitals that are formed after the mixing are called ‘Hybrid Orbitals’. Based on mixing hybridization is divided into $ 6 $ types, these are $ sp,\,s{p^2},s{p^3},s{p^3}d,s{p^3}{d^2},{d^2}s{p^3} $ .

Complete Step-by-step Answer:
In $ sp $ hybridization one $ s $ and one $ p $ orbital of the same atom combined together to form two new hybridized $ sp $ orbital of equal energy. The $ sp $ orbital has equal representation of $ s $ and $ p $ character i.e. they both are $ 50\% $ . $ sp $ hybridisation is also called diagonal hybridisation. The type of arrangement in molecules with $ sp $ hybridization is linear with a bond angle of $ {180^0} $ . Examples of molecules that show $ sp $ hybridization are $ BeC{l_2},\,{C_2}{H_2},Be{F_2} $ . The shape of $ sp $ orbital is;
In $ s{p^2} $ hybridization one $ s $ and two $ p $ orbital of the same atom combined together to form three new hybridized $ s{p^2} $ orbital of equal energy. The $ s{p^2} $ orbital does not have an equal representation of $ s $ and $ p $ character i.e. $ s $ character is $ 33.33\% $ and $ p $ character is $ 66.66\% $ . $ s{p^2} $ hybridisation is also called trigonal hybridisation. The type of arrangement in molecules with $ s{p^2} $ hybridization is Trigonal planar with a bond angle of $ {120^0} $ . Examples of molecules that show $ s{p^2} $ hybridization are $ $ B{F_3},\,{C_2}{H_4},B{H_3} $ $ . The shape of $ s{p^2} $ orbital is;
In $ s{p^3} $ hybridization one $ s $ and three $ p $ orbital of the same atom combined together to form four new hybridized $ s{p^3} $ orbital of equal energy. The $ s{p^3} $ orbital does not have an equal representation of $ s $ and $ p $ character i.e. $ s $ character is $ 25\% $ and $ p $ character is $ 75\% $ . The type of arrangement in molecules with $ s{p^3} $ hybridization is Tetrahedral with a bond angle of $ {109.5^0} $ . Examples of molecules that show $ s{p^3} $ hybridization are ethane and ammonia. The shape of $ s{p^3} $ orbital is;

Note:
Hybridization is the phenomenon in which atomic orbitals that have similar energy are combined together to make new atomic orbitals with different structures and properties. As the energy in the hybrid orbitals is generalized, the repulsion between them is less and has relatively more stability.