The First Law of Thermodynamics

Statement and Meaning of the First Law

In chemical thermodynamics, the first law of thermodynamics expresses the quantitative connection between heat, work, and energy. In its simplest form:

$$
\Delta U = Q + W
$$

Here:

• $U$ – internal energy of a system
• $\Delta U$ – change in internal energy between an initial and a final state
• $Q$ – heat supplied to the system (positive if added to the system)
• $W$ – work done on the system (positive if done on the system)

This is a conservation of energy statement applied to thermodynamic systems: energy can be transformed between different forms but cannot be created or destroyed.

For chemical purposes we often distinguish different kinds of work, and we very often work at (approximately) constant pressure, which leads to useful special forms of the first law.

Sign Convention and Types of Work

The first law needs a clear sign convention and a distinction of work types.

Sign Conventions

Two sign conventions are commonly used:

1. Chemistry convention (widely used in chemistry textbooks):
• $Q > 0$: heat flows into the system (system absorbs heat)
• $W > 0$: work is done on the system
• First law: $\Delta U = Q + W$
2. Physics convention (frequent in physics):
• $Q > 0$: heat flows into the system
• $W > 0$: work is done by the system
• First law is then written as:
  $$
  \Delta U = Q - W
  $$

In this course we use the chemistry convention: positive $W$ = work on the system.

$pV$-Work (Expansion and Compression)

For many chemical processes, the dominant type of work is pressure–volume work ($pV$-work), associated with volume changes of gases:

• Reversible $pV$-work:
  $$
  \mathrm{d}W_{\text{rev}} = -p_{\text{ext}} \,\mathrm{d}V
  $$
  where $p_{\text{ext}}$ is the (infinitesimally different) external pressure and $V$ is the system volume.

Using the chemistry sign convention:

• If the system expands ($\mathrm{d}V > 0$) against external pressure, it does work on the surroundings. Then $\mathrm{d}W_{\text{on sys}} = -p_{\text{ext}}\mathrm{d}V < 0$.
• If the system is compressed ($\mathrm{d}V < 0$), work is done on the system; $\mathrm{d}W_{\text{on sys}} > 0$.

For finite changes at constant external pressure $p_{\text{ext}}$:

$$
W = -p_{\text{ext}}\,\Delta V
$$

with $\Delta V = V_{\text{final}} - V_{\text{initial}}$.

Examples:

• Gas expansion in a piston during a chemical reaction
• Compression of a gas in a cylinder

Non-$pV$ Work

In many chemical and electrochemical processes there are additional forms of work:

• Electrical work in electrochemical cells (moving electrical charge against an electric potential difference)
• Surface work (e.g. creating new surface in interfacial systems)
• Mechanical (shaft) work (e.g. stirring via a motor)

These are often grouped as non-expansion work $W_{\text{non-}pV}$:

$$
W = W_{pV} + W_{\text{non-}pV}
$$

The first law then reads:

$$
\Delta U = Q + W_{pV} + W_{\text{non-}pV}
$$

For many simple systems (e.g. gases in a cylinder without electrical effects) we can neglect $W_{\text{non-}pV}$ and focus on $pV$-work.

Internal Energy as a State Function

The internal energy $U$ is a state function:

• It depends only on the state of the system (defined by variables such as $T$, $p$, $V$, composition),
• It does not depend on the path by which the system reached that state.

In contrast, both heat $Q$ and work $W$ are path-dependent quantities:

• They depend on the exact process connecting the same initial and final states.
• We do not write $Q$ or $W$ without a process description; they are not properties of a state.

Mathematically, for different processes $1$ and $2$ between the same two states $A$ and $B$:

• $U_A$ and $U_B$ are unique and independent of the path. So:
  $$
  \Delta U = U_B - U_A
  $$
  is the same for all paths.
• $Q_1 \neq Q_2$ and $W_1 \neq W_2$ are generally possible, but:
  $$
  \Delta U = Q_1 + W_1 = Q_2 + W_2
  $$

This is the essence of the first law: only the combination $Q + W$ is path-independent.

Work and Heat in Simple Processes

To use the first law in practice, we consider particular types of processes that occur frequently in chemistry, especially for ideal gases.

Isochoric Processes (Constant Volume)

Isochoric: volume $V$ is constant, so:

$$
\Delta V = 0 \quad\Rightarrow\quad W_{pV} = -p_{\text{ext}} \Delta V = 0
$$

If we neglect other work forms, $W=0$ and the first law simplifies to:

$$
\Delta U = Q_V
$$

where $Q_V$ is the heat exchanged at constant volume. Thus:

• The heat supplied at constant volume directly equals the change in internal energy.

This relation is frequently used when a reaction is carried out in a closed rigid container (e.g. a sealed bomb calorimeter).

Isobaric Processes (Constant Pressure)

Isobaric: pressure $p$ is constant. Many chemical processes occur (approximately) at constant external pressure, for example reactions open to the atmosphere.

For a constant external pressure $p$:

$$
W_{pV} = -p\,\Delta V
$$

Then, with $W = W_{pV}$:

$$
\Delta U = Q_p - p\,\Delta V
$$

Here $Q_p$ is the heat exchanged at constant pressure.

This form connects $\Delta U$ to $Q_p$ and volume change. It provides the bridge to the enthalpy function, which is treated separately.

Reversible and Irreversible Processes in the Context of the First Law

The first law itself does not distinguish between reversible and irreversible processes; it holds for both. However, the expressions for $W$ and $Q$ for a given $\Delta U$ typically differ between reversible and irreversible paths.

• Reversible expansion/compression: system pressure is always in infinitesimal balance with external pressure.
• The $pV$-work is maximal (for expansion) or minimal (for compression), in magnitude.
• Work can be calculated by integrating:
  $$
  W_{\text{rev}} = -\int_{V_i}^{V_f} p(V)\,\mathrm{d}V
  $$
• Irreversible expansion/compression: sudden expansion, friction, finite pressure differences.
• $p_{\text{ext}}$ is constant and differs significantly from system pressure.
• $W_{\text{irr}} = - p_{\text{ext}}\Delta V$ generally has a smaller magnitude (for the same $\Delta V$) than $W_{\text{rev}}$.

For a given initial and final state, therefore:

• $\Delta U$ is the same (state function),
• $Q_{\text{rev}}$ and $W_{\text{rev}}$ differ from $Q_{\text{irr}}$ and $W_{\text{irr}}$, but:
  $$
  \Delta U = Q_{\text{rev}} + W_{\text{rev}} = Q_{\text{irr}} + W_{\text{irr}}
  $$

The concept of reversibility becomes essential when introducing entropy and the second law.

Application to Ideal Gases

For an ideal gas, the internal energy depends only on the temperature $T$ and the amount of substance $n$, not on volume or pressure:

$$
U = U(T, n) \quad\Rightarrow\quad \Delta U = n C_V \Delta T
$$

where:

• $C_V$ – molar heat capacity at constant volume,
• $\Delta T$ – temperature change.

Thus:

• If an ideal gas is heated from $T_1$ to $T_2$ at constant volume:
  $$
  \Delta U = n C_V (T_2 - T_1) = Q_V
  $$
• At constant pressure with temperature change $\Delta T$:
• We still have $\Delta U = n C_V \Delta T$,
• But now $W_{pV} = -p\Delta V$ is not zero, and $Q_p \neq Q_V$.

The linear relationship $\Delta U \propto \Delta T$ for ideal gases simplifies many calculations in chemical thermodynamics, for example in calorimetry and reaction energetics.

First Law for Closed and Open Systems

So far we have assumed a closed system (no matter exchange). In many chemical processes (e.g. flow reactors, living cells) matter does enter or leave the system; these are open systems.

Closed Systems

• No exchange of matter across the system boundary.
• Energy exchange only via heat and work.
• First law:
  $$
  \Delta U = Q + W
  $$

This form is most often used for laboratory-scale reactions in sealed vessels, pistons, or calorimeters.

Open Systems (Overview)

In an open system, there is in addition:

• Energy transport with matter (enthalpy of flowing streams),
• Possible work terms associated with flow.

The full energy balance must then include:

• Heat,
• Work,
• Energy carried into and out of the system by flowing matter.

In many introductory chemical thermodynamics treatments, one first focuses on closed systems and later extends the first law to open systems, particularly in the context of chemical engineering and biological systems.

Practical Relevance in Chemistry

The first law underlies:

• Calorimetry – determining reaction energies from measured heat effects:
• At constant volume: $Q_V = \Delta U_{\text{reaction}}$ for the system
• At constant pressure: $Q_p$ connected to enthalpy change
• Energy balances in industrial processes:
• Heating and cooling demands
• Work requirements for compression, expansion, and pumping
• Understanding exothermic and endothermic reactions:
• Exothermic: reaction releases heat to surroundings (at given conditions)
• Endothermic: reaction absorbs heat from surroundings

In all these cases, the first law provides the quantitative framework to relate heat measured experimentally to changes in internal energy and work performed or required.

Summary of Key Relations

For the use of the first law in chemical thermodynamics:

• General form (chemistry convention):
  $$
  \Delta U = Q + W
  $$
• For $pV$-work at constant external pressure:
  $$
  W_{pV} = -p_{\text{ext}}\,\Delta V
  $$
• Constant volume (closed system, only $pV$ work):
  $$
  W = 0,\quad \Delta U = Q_V
  $$
• Constant pressure (closed system, only $pV$ work):
  $$
  \Delta U = Q_p - p\,\Delta V
  $$
• Ideal gas:
  $$
  \Delta U = n C_V \Delta T
  $$

These relations form the working toolkit derived from the first law for many basic thermodynamic and chemical calculations.