The molar volume of liquid benzene (density = 0.877 g mL^-1) increases by a factor of 2750 as it vaporises at 20^oC and that of liquid toluene (density = 0.867 g mL^-1) increases by a factor of 7720 at 20^oC. A solution of benzene and toluene at 20^oC has vapour pressure of 46.0 Torr. Find the mole fraction of benzene in the vapour above the solution.

The standard reduction potential for Cu²⁺|Cu is +0.34 V. Calculate the reduction potential at pH = 14 for the above couple. K_sp of Cu(OH)₂ is 1.0 $$\times$$ 10^-19

The ionisation constant of $$NH_4^+$$ in water is 5.6 $$\times$$ 10^-10 at 25^oC. The rate constant for the reaction of $$NH_4^+$$ and $$OH^-$$ to form NH₃ and H₂O at 25^oC is 3.4 $$\times$$ 10¹⁰ L mol^-1s^-1. Calculate the rate constant for proton transfer from water to NH₃.

For $$n>0,$$ $$\int_0^{2\pi } {{{x{{\sin }^{2n}}x} \over {{{\sin }^{2n}}x + {{\cos }^{2n}}x}}} dx = $$