Work and heat

First Law of Thermodynamics

Δ U = Q - W

$Δ U$ : change in internal energy
$Q$ : heat absorbed by system (+) / lost by system (−)
$W$ : work done by system (+) / done on system (−)

Thermodynamic Processes

Process	Condition	Simplification
Isovolumetric	$Δ V = 0$	$W = 0$ , so $Δ U = Q$
Isothermal	$Δ T = 0$	$Δ U = 0$ , so $Q = W$
Adiabatic	$Q = 0$	$Δ U = - W$
Isolated	$Q = 0$ , $W = 0$	$Δ U = 0$

Sign Conventions

$Q > 0$ : heat flows into system
$W > 0$ : work done by system (expansion)
$W < 0$ : work done on system (compression)

Microscopic Picture

Internal energy as a sum over microstates: $U = \sum_{i} p_{i} E_{i}$

Total differential:

d U = \sum_{i} E_{i} d p_{i} + \sum_{i} p_{i} d E_{i}

d U = đ q + đ w

$đ q = \sum_{i} E_{i} d p_{i}$ — heat: change probabilities $p_{i}$ , keep energy levels $E_{i}$ fixed
$đ w = \sum_{i} p_{i} d E_{i}$ — work: change energy levels $E_{i}$ (e.g. by compression), keep $p_{i}$ fixed

$đ q$ and $đ w$ are inexact differentials — unlike $d U$ , their integrals depend on the path taken between states, not just the endpoints. Heat and work are not state functions; only their sum is.