What is the second law of thermodynamics? Give its mathematical

expression. For a reaction, both DH and DS are positive. Predict thecondition for the reaction to be spontaneous. What is the significance ofTDS in the expression DG = DH -TDS ?

According to second law of thermodynamics, it is impossible to construct a heat-engine which converts all the energy supplied to it in to work. In other words, heat engines with 100 % efficiency cannot be constructed.
Another form also states that entropy always increases in case of irreversible processes and remains constant for a reversible process. It provides a direction in which a thermodynamic process proceeds,i.e., the direction in which entropy increases. The mathematical equation for second law of thermodynamics is:
S_universe = S_system + S _surrounding $\ge$0
where  ΔS_universe  equals zero for a truly reversible process and is greater than zero for an irreversible process. 

• The change in Gibbs energy for the system ΔG _sys, is given by

At constant T, ΔT = 0

Or, 

• ΔG has the units of energy (KJmol^-1).
• ΔG is related to reaction spontaneity. It gives the criteria for spontaneity at constant pressure and temperature.
• For spontaneous process,

∴− ΔG > 0

Or, ΔG = ΔH − TΔS < 0

• If ΔG < 0 (i.e., negative), the process is spontaneous.
• If ΔG > 0 (i.e., positive), the process is non-spontaneous.
• When a reaction has a positive enthalpy change and positive entropy change, it can be spontaneous if TΔS is large enough to outweigh ΔH.