First Law of Thermodynamics

1. Introduction

Thermodynamics is the branch of physics that deals with the transformation of heat into other forms of energy (and the reverse), and with the relationships between heat, work, internal energy, and the observable properties (pressure, volume, temperature) of a system. The first law of thermodynamics is simply the principle of conservation of energy applied to heat processes: it keeps a strict account of the heat supplied to a system, the change in its internal energy, and the work it does.

This chapter introduces the thermodynamic system, internal energy, and work done during a volume change, states and applies the first law, defines the two molar specific heats of a gas, and analyses the four standard thermodynamic processes (isothermal, isobaric, isochoric, adiabatic).

2. Thermodynamic System, Surroundings and Boundary

A thermodynamic system is a definite quantity of matter (for example a fixed mass of gas) that is chosen for study and is separated from its surroundings by a real or imaginary boundary. Everything outside the boundary that can exchange energy or matter with the system is the surroundings.

Systems are classified by what they can exchange with the surroundings:

• Open system: exchanges both matter and energy with the surroundings. Example: water boiling in an open pot.
• Closed system: exchanges only energy (heat or work), not matter. Example: a fixed mass of gas sealed in a cylinder with a movable piston.
• Isolated system: exchanges neither matter nor energy. Example: hot liquid in an ideal thermos flask (approximately).

A thermodynamic state of the system is fixed by its state variables (pressure $P$, volume $V$, temperature $T$). A quantity whose value depends only on the current state, not on the path taken to reach it, is called a state function (for example $P$, $V$, $T$, and internal energy $U$).

3. Internal Energy and Latent Heat

3.1 Internal energy

The internal energy $U$ of a system is the total microscopic energy contained in it: the sum of the kinetic energies of all its molecules (due to translation, rotation and vibration) and the potential energies arising from intermolecular forces.

• Internal energy is a state function: it depends only on the present state of the system, not on how that state was reached.
• For an ideal gas there are no intermolecular forces, so the internal energy depends only on temperature:

$\Delta U = nC_v\,\Delta T$

This result holds for an ideal gas in any process, not just at constant volume, because $U$ depends on $T$ alone.

3.2 Latent heat

Latent heat is the heat absorbed or released by a substance during a change of phase (state) at constant temperature. Because the temperature does not change, this heat goes entirely into changing the molecular arrangement and the intermolecular potential energy, that is, into changing the internal energy configuration rather than the kinetic energy of the molecules. The heat needed to change the phase of mass $m$ is:

$Q = mL$

where $L$ is the specific latent heat (of fusion or of vaporization).

4. Work Done by a Gas and the Indicator (P-V) Diagram

Consider a gas enclosed in a cylinder by a frictionless piston of cross-sectional area $A$. If the gas pressure is $P$, the outward force on the piston is $F = PA$. When the gas expands and pushes the piston out through a small distance $dx$, the small work done by the gas is:

$dW = F\,dx = PA\,dx = P\,dV$

since $A\,dx = dV$ is the small increase in volume. For a finite change of volume from $V_1$ to $V_2$:

$W = \int_{V_1}^{V_2} P\,dV$

Indicator (P-V) diagram. A graph of pressure against volume is called an indicator diagram. The integral above is exactly the area under the P-V curve between $V_1$ and $V_2$:

• In an expansion ($V_2 > V_1$) the work done by the gas is positive.
• In a compression ($V_2 < V_1$) the work done by the gas is negative (work is done on the gas).
• The work done depends on the path taken on the P-V diagram, so work is not a state function.