Heat

With the sign convention for energy transfer now fixed, we can define heat precisely. Consider a glass of water on a table (the system), at temperature $T_S$, with the table and the rest of the room as its environment, at temperature $T_E$. If $T_S \ne T_E$, the water's temperature will keep changing until the water and its surroundings reach a common temperature — thermal equilibrium is re-established between the 'system' and the 'environment'. (If the environment is very large — imagine a huge room — its own temperature change may be too small to notice, but it will not be exactly zero.)

This change in temperature is driven by a transfer of internal energy between the system and its environment. We define heat, denoted $Q$, as exactly this: the energy transferred between a system and its environment purely because of a temperature difference between the two.

Three distinct cases follow directly from the sign convention:

• If $T_S < T_E$ (the environment is hotter than the system), energy flows into the system, so the system gains energy and $Q$ is positive.
• If $T_S > T_E$ (the system is hotter than the environment), energy flows out of the system, so the system loses energy and $Q$ is negative.
• If $T_S = T_E$ (system and environment are already in thermal equilibrium), there is no net transfer of energy at all, so $Q = 0$. …