A mass of a liquid, \(m\), at temperature, \(T_{1}\), is mixed with an equal mass of the same liquid at temperature, \(T_{2}\). The system is thermally insulated. Show that the change of entropy of the Universe is

\[2 m c_{p} \ln \left(\frac{\left(T_{1}+T_{2}\right) / 2}{\sqrt{T_{1} T_{2}}}\right)\]

and prove that this is necessarily positive.

[Hint: Do not specifically consider the mixing of the fluids.]