Energy Accounting and Directionality in Thermodynamics (AP Physics 2, Unit 1)

First Law of Thermodynamics

Energy-conservation rule for thermal systems: changes in a system’s internal energy come from heat transfer and work interactions.

First Law (AP Physics sign convention)

ΔU = Q − W, where Q is heat added to the system and W is work done by the system.

Internal Energy (U)

Microscopic energy stored in a system (particle kinetic + molecular potential energy); problems usually focus on the change ΔU rather than U itself.

Heat (Q)

Energy transferred because of a temperature difference between a system and its surroundings.

Heat sign convention

Q > 0 means heat flows into the system; Q < 0 means heat flows out of the system.

Work (W) in the First Law

Energy transferred by a force through a distance at the system boundary (often via expansion/compression).

Work sign convention (AP: ΔU = Q − W)

$W > 0$ when the system does work on the surroundings (system loses energy via work); $W < 0$ when surroundings do work on the system (system gains energy via work).

State Function

A quantity that depends only on the current state (not the path); for a given start and end state, its change is path-independent.

Path-Dependent Quantity

A quantity whose value depends on the process path between states; heat Q and work W can differ for different paths even if ΔU is the same.

Quasi-static Work (PV work)

For a slow process with well-defined pressure, the work done by a gas equals the area under the curve on a P–V graph: W = \textstyle{\bigg\\tiny{\bigg\\begin{matrix}\times\bigg\\text{dV}\text{dT}\backslash\text{H}\bigg\text{dV}\bigg\text{dT}\bigg)}\bigg\text{sweeping up an arc}\bigg\bigg}(added done must not);$\backslash$

Constant-Pressure Work

If pressure is constant, work done by the gas is W = PΔV.

Expansion vs. Compression (work sign)

Expansion ($ΔV > 0$) gives $W > 0$ (system does work); compression ($ΔV < 0$) gives $W < 0$ (work done on system).

Isochoric Process

Constant-volume process (ΔV = 0), so W = 0 and the First Law reduces to ΔU = Q.

Adiabatic Process

No heat transfer (Q = 0), so ΔU = −W; internal energy changes only via work.

Cyclic Process

A process that returns to the initial state; over a full cycle $\triangle U_{cycle} = 0$, so $Q_{net} = W_{net}$.

Second Law of Thermodynamics

Adds directionality to energy transfers: not all energy-conserving processes happen spontaneously; natural processes tend toward greater overall energy dispersal (entropy increase for the universe).

Kelvin–Planck Statement

No cyclic device can take heat from a single reservoir and convert it entirely into work (some waste heat must be rejected).

Clausius Statement

Heat does not spontaneously flow from a colder object to a hotter object.

Heat Engine

A cyclic device that absorbs heat $Q_H$ from a hot reservoir, rejects heat $Q_C$ to a cold reservoir, and produces net work output.

Heat Engine Work Output

Over a cycle ($\triangle U = 0$): $W_{out} = Q_{H} - Q_{C}$.

Thermal Efficiency (e)

Fraction of input heat converted to work: $e = \frac{W_{out}}{Q_H} = 1 - \frac{Q_C}{Q_H}$; for real engines $e < 1$.

Refrigerator (energy relation)

A device that uses work input to move heat from cold to hot; over a cycle: $Q_H = Q_C + W_{in}$.

Coefficient of Performance (COP)

Performance measure for heat movers: refrigerator COP $K_R = \frac{Q_C}{W_{in}}$; heat pump COP $K_{HP} = \frac{Q_H}{W_{in}}$ (can be $> 1$).

Carnot Engine Efficiency

Maximum possible efficiency for any engine operating between $T_H$ and $T_C$ (in kelvins): $e_{Carnot} = 1 - \frac{T_C}{T_H}$.

Entropy (S) and Entropy Change

State function measuring energy dispersal/microstate availability; for reversible heat transfer at constant temperature: $\triangle S = \frac{Q_{rev}}{T}$ (T in kelvins).