In the discussion following Equation $20.43$ it was said that $Q_1=-Q_2$. Prove that this is so.

By virtue of its temperature, thermal energy is the internal energy existing in a system in thermodynamic equilibrium. Thermal energy is more difficult to convert to productive work than energy from systems that are not in thermodynamic equilibrium.

The quantity of thermal energy that is transferred from one system to another is referred to as heat. The first system has a heating element.

$Q_1=\Delta E_1=E_{1\mathrm{f}}-E_{1\mathrm{i}}$---------(1)

The second system has a heat

$Q_2=\Delta E_2=E_{2\mathrm{f}}-E_{2\mathrm{i}}$--------(2)

If we have two systems, each one starts with thermal energy at a different temperature. When the two systems are coupled, they interact thermally until thermal equilibrium is reached. The initial thermal energy and the final thermal energy are the same since the energy is maintained during the contact.

$E_i=E_f$

The initial energy is $E_i=E_{1i}+E_{2i}$and the final energy is$E_f=E_{1f}+E_{2f}$. Equal both energies to get

$E_{1i}+E_{2i}=E_{1f}+E_{2f}$

$E_{1f}-E_{1i}=E_{2i}-E_{2f}$

$E_{1f}-E_{1i}=-\left(E_{2f}-E_{2i}\right)$-----------(3)

From equations (1) and $\left(2\right)$, we can rewrite equation $\left(3\right)$in the form

$Q_1=-Q_2$

In order to verify that the final and starting energies are equal, you must apply the conservation law of energy. $Q_1=-Q_2$.