2 Energy, Heat, and Work

Energy in a thermodynamic system

Energy is a property of a system. Its total energy can include internal energy, macroscopic kinetic energy, and gravitational potential energy; chemical, electrical, magnetic, or nuclear energy may also matter, depending on the system.

For a closed system, a useful accounting is

where $U$ is internal energy, $KE$ is kinetic energy, and $PE$ is potential energy. Include only the energy forms relevant to the situation.

Energy may cross a system boundary as heat or work. In an open system, energy can also enter or leave with matter flowing across the boundary. Heat and work describe transfers during a process; they are not properties stored in the system.

Heat and work as transfers

Heat is energy transferred across a boundary because of a temperature difference. It flows spontaneously from higher temperature toward lower temperature. Conduction, convection, and thermal radiation are common ways heat is transferred. After transfer, the energy is part of the receiving system’s energy, rather than heat stored in that system.

Work is energy transferred across a boundary through an organized interaction other than a temperature difference. Whether a transfer counts as heat or work depends on how energy crosses the chosen boundary, not simply on whether the energy eventually warms the system.

Sign convention and energy accounting

This guide uses the convention that heat into the system is positive and work done by the system is positive. Under this convention, the first law of thermodynamics for a closed system is

where $Q$ is net heat transferred into the system and $W$ is net work done by the system. Heat entering adds energy, while work done by the system transfers energy out. Heat leaving has $Q < 0$; work done on the system has $W < 0$. If changes in kinetic and potential energy are negligible, $\Delta E$ is approximately $\Delta U$.

Some fields and textbooks instead define work as positive when it is done on the system, giving $\Delta E = Q + W_{\text{on}}$. The conventions are equivalent when used consistently, so check how a problem defines work.

For example, a gas loses $30\,\text{J}$ of heat while being compressed, and $80\,\text{J}$ of work is done on it. Under the convention used here, $Q = -30\,\text{J}$ and $W = -80\,\text{J}$, so

The gas’s total energy increases by $50\,\text{J}$.

Common modes of work

Work can cross a system boundary in several ways. Boundary work is one common mode, but it is not the only kind of work.

• Boundary work: A moving boundary, such as a piston, transfers energy as the system expands or contracts. For a quasistatic process,

  $$W_{\text{by}} = \int_{V_1}^{V_2} p\,dV.$$

  Here, $p$ is the system pressure during the process. Expansion gives positive work by the system; compression gives negative work by the system.

• Shaft work: A rotating shaft transfers energy through torque and rotation, as in a turbine, compressor, or paddle wheel.

• Electrical work: Energy crosses the boundary through electrical forces and current, as when a motor or heating element is connected to a system.

• Other mechanical work: A moving force can transfer energy through a boundary, for example by stretching a spring or raising a weight.

For any mode, identify the system boundary, decide whether energy transfer is into or out of the system, and apply the stated sign convention.