Internal energy work and heat

Internal energy

Internal energy is the total microscopic energy contained in a system. It includes molecular kinetic energy, intermolecular potential energy, chemical energy, and other microscopic forms. It is usually written as $U$.

For an ideal monatomic gas, internal energy depends only on temperature:

U=rac{3}{2}nRT

where $n$ is number of moles, $R$ is the gas constant, and $T$ is absolute temperature.

Real substances can have more complicated internal energy because molecules rotate, vibrate, interact, and change phase.

Heat and work

Heat and work are two ways energy crosses a system boundary. Heat, written $Q$, is energy transferred because of temperature difference. Work, written $W$, is energy transferred by macroscopic forces acting through displacements.

Neither heat nor work is stored in a system. A system stores internal energy. Heat and work describe transfer processes.

The first law

A common sign convention writes the first law as

$Delta U=Q-W$

Here $Q$ is heat added to the system, and $W$ is work done by the system on the surroundings. If heat enters the system, $Q>0$. If the system expands and does work on the outside world, $W>0$.

Some books use

$Delta U=Q+W_{on}$

where $W_{on}$ is work done on the system. Both conventions are valid if used consistently.

Pressure-volume work

For a gas in a piston, expansion work is

$dW=P,dV$

For a finite process,

$W=int_{V_i}^{V_f}P,dV$

This work is the area under a pressure-volume curve. If the gas expands, $dV>0$ and work done by the gas is positive. If compressed, $dV<0$ and work done by the gas is negative.

Path dependence

Work and heat depend on the path taken between states. The same initial and final states can be connected by different processes with different amounts of heat and work. Internal energy change, however, depends only on the initial and final states.

This is why $U$ is a state function while $Q$ and $W$ are process quantities.

Example: heating at constant volume

If a gas is heated in a rigid container, volume does not change, so

$W=0$

The first law becomes

$Delta U=Q$

All added heat increases internal energy.

Example: expansion with no heat

If a gas expands adiabatically, no heat is exchanged:

$Q=0$

Then

$Delta U=-W$

If the gas does work during expansion, its internal energy decreases. For an ideal gas, this means temperature decreases.

Microscopic interpretation

Adding heat to a system usually increases microscopic energy. Doing work on a system can also increase internal energy, such as compressing a gas or stirring a liquid. The first law does not say temperature always rises when heat is added, because energy may also leave as work or go into phase change.

The big idea

The first law of thermodynamics is conservation of energy applied to thermal systems. Internal energy is stored microscopic energy. Heat and work are energy transfers. The equation $Delta U=Q-W$ organizes thermal processes and makes clear that energy can change form but not disappear.