3.11.2.1 - First Law of Thermodynamics

Systems and internal energy

A system is the matter we choose to study. Its boundary separates it from its surroundings. A sealed gas cylinder is a closed system: no matter crosses the boundary, although a piston may move and energy may cross. A pipe carrying gas is an open system because matter enters and leaves.

[DIAGRAM: asset_name: C.2.1 - First Law of Thermodynamics - Diagram 1; asset_slug: C.2.1 - First Law of Thermodynamics - Diagram 1; caption: An open system exchanges matter, a closed system keeps the same particles even when its boundary moves.; description: An open system exchanges matter, a closed system keeps the same particles even when its boundary moves.]

Internal energy, $U$, is the sum of the randomly distributed kinetic and potential energies of the particles. It excludes the kinetic energy of the whole cylinder moving through a room. Internal energy is a state function: returning to the same thermodynamic state returns the system to the same internal energy.

For a fixed amount of ideal gas, internal energy depends only on temperature. For substances in general, temperature alone does not determine internal energy: a change of phase can change potential energy at constant temperature.

The first-law energy balance

Heating and work are ways of transferring energy across the boundary; they are not separate stores of energy inside the gas. For a closed system with no relevant change in bulk kinetic or gravitational energy, conservation gives:

$Q$ is energy transferred to the system by heating, $\Delta U$ is its increase in internal energy and $W$ is work done by the system. All three are in joules. This lesson considers pressure–volume work; if electrical work or stirring is supplied, it must also be included consistently.

If a gas receives $500\,\mathrm{J}$ by heating and transfers $200\,\mathrm{J}$ to a piston, its internal energy increases by $500-200=300\,\mathrm{J}$. Heating need not raise the temperature by itself: compare the heat input with the work output.

Assigning signs before calculating

Assign signs from the direction of each energy transfer before substituting numbers. The work convention must stay the same throughout a calculation.

For expansion against a positive external pressure, $W>0$. For compression, $W<0$. Expansion into a vacuum is an important exception: there is no resisting force, so no work is transferred.

Worked compression: $1200\,\mathrm{J}$ of work is done on a gas while $400\,\mathrm{J}$ leaves by cooling. Thus $W=-1200\,\mathrm{J}$, $Q=-400\,\mathrm{J}$ and

The gas gains internal energy despite losing energy by heating. A book using work on the gas writes $\Delta U=Q+W_{\mathrm{on}}$, with $W_{\mathrm{on}}=-W$; these are the same physics.