Why does the entropy of a substance increase with an increase in temperature?

One mole of an ideal gas expands adiabatically and reversibly from its initial pressure to half of its initial pressure. What will be the value of $\Delta S$ in $J \ K^{-1} \ mol^{-1}$? $[\ln 2 = 0.693$ and $R = 8.314 \ J \ mol^{-1} \ K^{-1}]$

Calculate the entropy change in the surroundings when $1.00 \ mol$ of $H_2O_{(l)}$ is formed under standard conditions. Given: $\Delta_f H^{\theta} = -286 \ kJ \ mol^{-1}$.

Given $\Delta _rS^o = -266 \ J \ K^{-1}$ and the listed $S_m^o$ values,calculate $S^o$ for $Fe_3O_{4(s)}$ for the reaction: $4 Fe_3O_{4(s)} + O_{2(g)} \to 6 Fe_2O_{3(s)}$. Given $S_m^o(Fe_2O_3) = 87 \ J \ K^{-1} \ mol^{-1}$ and $S_m^o(O_2) = 205 \ J \ K^{-1} \ mol^{-1}$.

Calculate the entropy change when $1 \, \text{mole}$ of ice is converted into water at $0 \, ^\circ\text{C}$. (The enthalpy change for the conversion of ice to liquid at $0 \, ^\circ\text{C}$ is $6.0 \, \text{kJ} \, \text{mol}^{-1}$).

Statement-$1$: For a process at equilibrium,the total entropy is maximum and the change in total entropy is zero.
Statement-$2$: For a reversible process,if the same amount of heat is supplied to a system at two different temperatures,then the entropy change is less at a lower temperature.