Internal energy — common mistakes

Pay meticulous attention to the sign conventions for Q and W! A common mistake is getting the signs wrong, especially for work done by versus on the system. Always ask: Is energy entering (positive) or leaving (negative) the system? Is work being done on (positive) or by (negative) the system?

Heat is the transfer of thermal energy between systems due to a temperature difference, while internal energy is the total energy stored within a system's particles. Internal energy is a property of the system, whereas heat is a process of energy transfer.

Yes, this occurs during phase changes (e.g., melting ice or boiling water). Energy added as latent heat increases the potential energy between particles (breaking bonds), even though the average kinetic energy and thus temperature remain constant. This means $\Delta U$ is positive even if $\Delta T$ is zero.

'Random' emphasises that we're considering the microscopic, disordered motion and interactions of individual particles, not the macroscopic kinetic energy of the system as a whole (e.g., a moving car) or its overall gravitational potential energy.

For an ideal gas undergoing an isothermal process, the change in internal energy ($\Delta U$) is zero. According to the First Law of Thermodynamics ($\Delta U = Q + W$), this means $0 = Q + W$, or$Q = -W$. Any heat added to the system must be completely converted into work done by the system, and vice versa.