For the system to reach thermal equilibrium, the block of material must transfer energy away from it at the same rate that it receives energy. This is because thermal equilibrium is achieved when the temperature of all components in the system are equal. Therefore, the rate of energy transfer into the material must equal the rate of energy transfer out of the material. If the material were gaining energy faster than it lost, its temperature would rise. Conversely, if it were losing energy faster than it gained, its temperature would fall. The system will continue to exchange energy until the temperature difference between the material and the water is eliminated, resulting in a constant temperature for the entire system.

Thermal radiation is a form of electromagnetic radiation, specifically infrared radiation. The energy of electromagnetic radiation is directly proportional to its frequency. Higher temperatures mean that the atoms and molecules within an object have greater kinetic energy and vibrate more vigorously. This increased vibration leads to the emission of more photons with higher energy and therefore a greater intensity of thermal radiation. The relationship between energy (E), frequency (f), and Planck's constant (h) is given by E = hf. Since higher temperatures result in higher vibrational frequencies and thus higher energy photons, more thermal radiation is emitted.

A thermometer works by exploiting the principle of thermal equilibrium. It typically consists of a glass tube containing a liquid, often mercury or alcohol. When the thermometer is placed in contact with a substance, heat is transferred between the two. If the substance is hotter than the liquid in the thermometer, heat flows from the substance to the liquid until they reach the same temperature. Conversely, if the substance is colder, heat flows from the liquid to the substance until they reach thermal equilibrium. The key is that the rate of heat transfer into the liquid equals the rate of heat transfer out of the liquid. The volume of the liquid expands or contracts with changes in temperature. This expansion or contraction is mechanically linked to a scale, allowing us to read the temperature. The thermometer achieves thermal equilibrium by allowing heat to flow freely between itself and the substance until their temperatures are equal.