Table of Contents
Direction of Physical and Chemical Processes
The first law of thermodynamics describes the conservation of energy, but it does not say anything about the direction in which processes occur. Many processes are energetically possible in both directions, yet in reality they proceed only one way under given conditions.
Examples:
- A hot metal block placed in cold water cools down; the water warms up. The reverse – the block getting hotter and water colder without external influence – is never observed.
- A gas released into a vacuum spreads out to fill the whole container. It never spontaneously returns to occupy only one half of the container.
- Two gases in separate compartments mix when the separating wall is removed. They never separate again on their own.
The second law of thermodynamics provides the principle that distinguishes between possible and impossible directions of processes and introduces a new state function: entropy.
Statements of the Second Law
There are many equivalent ways to express the second law. Two classical, practical formulations relate to heat engines and refrigerators.
Kelvin–Planck Statement
It is impossible to construct a cyclic process whose only result is:
- the extraction of heat from a single heat reservoir and
- the complete conversion of this heat into work.
In other words:
No heat engine can operate with 100% efficiency when working between a single heat source and itself; part of the absorbed heat must be rejected to a cooler reservoir.
This distinguishes heat from work: while work can be completely converted to heat, heat cannot be completely converted into work in a cyclic process.
Clausius Statement
It is impossible to construct a cyclic process whose only result is:
- the transfer of heat from a colder body to a hotter body.
That is, heat will not spontaneously flow from cold to hot without external work being performed (e.g. by a refrigerator compressor).
Equivalence of the Statements
While these two statements sound different, they are logically equivalent. A device violating one statement could be used to construct a device violating the other. Thus, if one formulation is true, the other must also be true.
Together, these formulations capture an essential observation: natural processes are irreversible in practice and have a preferred direction.
Entropy as a State Function
To make the second law quantitative, the concept of entropy $S$ is introduced. It is a state function of a thermodynamic system, like internal energy $U$, and depends only on the current state, not on the path taken to reach it.
Definition for Reversible Processes
For a reversible (idealized, infinitely slow) process, the infinitesimal change in entropy $dS$ is defined by
$$
dS = \frac{\delta Q_\mathrm{rev}}{T}
$$
where
- $\delta Q_\mathrm{rev}$ is the infinitesimal amount of reversible heat absorbed by the system,
- $T$ is the absolute temperature (in kelvin) at which the heat exchange takes place.
Because $S$ is a state function, its change between two states 1 and 2 is
$$
\Delta S = S_2 - S_1 = \int_{1}^{2} \frac{\delta Q_\mathrm{rev}}{T}
$$
Even if the real (irreversible) process between 1 and 2 is complicated, $\Delta S$ can be calculated by imagining any convenient reversible path connecting the same initial and final states.
Units of Entropy
From the definition, the SI unit of entropy is
$$
\mathrm{J \, K^{-1}}
$$
since heat is measured in joules and temperature in kelvin.
Entropy Changes in Simple Processes
In this chapter we focus on expressions for entropy changes that are frequently used in chemistry. (The detailed derivations and examples are treated elsewhere.)
Isothermal Entropy Change for Ideal Gases
For an ideal gas undergoing a reversible isothermal (constant $T$) expansion or compression from volume $V_1$ to $V_2$:
$$
\Delta S = nR \ln \frac{V_2}{V_1}
$$
Equivalently, using pressures for an isothermal ideal gas process:
$$
\Delta S = - nR \ln \frac{p_2}{p_1}
$$
where
- $n$ is the amount of substance in mol,
- $R$ is the universal gas constant.
If $V_2 > V_1$ (expansion), then $\Delta S > 0$. If $V_2 < V_1$ (compression), then $\Delta S < 0$.
Entropy Change with Temperature at Constant Heat Capacity
For a substance with constant molar heat capacity $C_p$ (approximation) heated reversibly from $T_1$ to $T_2$ at constant pressure:
$$
\Delta S = n C_p \ln \frac{T_2}{T_1}
$$
At constant volume, $C_p$ is replaced by $C_V$:
$$
\Delta S = n C_V \ln \frac{T_2}{T_1}
$$
Entropy Change in Phase Transitions
For a reversible phase transition at constant temperature and pressure (e.g. melting, vaporization) the entropy change is given by
$$
\Delta S_\mathrm{phase} = \frac{\Delta H_\mathrm{phase}}{T_\mathrm{transition}}
$$
where $\Delta H_\mathrm{phase}$ is the molar enthalpy change of the phase transition (e.g. enthalpy of fusion or vaporization).
Typical features:
- Melting: $\Delta S_\mathrm{fusion} > 0$ (solid $\to$ liquid)
- Freezing: $\Delta S_\mathrm{freezing} < 0$
- Vaporization: $\Delta S_\mathrm{vap} > 0$ (often large, due to big increase in freedom of molecules)
Second Law in Terms of Entropy
The second law can be expressed in a general, quantitative way using entropy and the concept of the universe (system + surroundings).
For any real (spontaneous) process:
$$
\Delta S_\mathrm{universe} = \Delta S_\mathrm{system} + \Delta S_\mathrm{surroundings} \ge 0
$$
- If $\Delta S_\mathrm{universe} > 0$, the process is spontaneous in the given direction.
- If $\Delta S_\mathrm{universe} = 0$, the process is reversible (ideal limit).
- If $\Delta S_\mathrm{universe} < 0$, the process in that direction is impossible; the reverse would be spontaneous.
This formulation applies equally to physical and chemical processes and underlies the later definition of Gibbs free energy for processes at constant temperature and pressure.
Entropy Change of the Surroundings (Isothermal, Isobaric)
For processes in chemistry, it is common to consider systems at constant external pressure and temperature (e.g. reactions in calorimeters coupled to a large thermal reservoir).
When a process at constant external pressure $p$ exchanges heat $Q_p$ with a large surroundings at temperature $T$, the entropy change of the surroundings is approximated by
$$
\Delta S_\mathrm{surroundings} = -\,\frac{Q_p}{T} = -\,\frac{\Delta H_\mathrm{system}}{T}
$$
where $\Delta H_\mathrm{system}$ is the enthalpy change of the system. This relation connects entropy changes of surroundings directly with enthalpy changes of the process.
Irreversibility and Entropy Production
Real processes in nature are irreversible. Irreversibility is always accompanied by entropy production.
For a given system, we can write:
$$
\Delta S_\mathrm{system} = \Delta S_\mathrm{e} + \Delta S_\mathrm{i}
$$
where
- $\Delta S_\mathrm{e}$ is entropy exchange with the environment (e.g. by heat transfer),
- $\Delta S_\mathrm{i}$ is entropy produced inside the system due to irreversibilities.
The second law requires:
$$
\Delta S_\mathrm{i} \ge 0
$$
- For a reversible process: $\Delta S_\mathrm{i} = 0$
- For an irreversible process: $\Delta S_\mathrm{i} > 0$
Sources of internal entropy production include:
- Heat flow through a finite temperature difference
- Unrestrained expansion (free expansion) of a gas
- Mixing of different gases or solutions
- Friction, viscosity, and inelastic deformation
- Chemical reactions proceeding towards equilibrium
The concept of entropy production is fundamental in understanding why ideal reversible processes represent theoretical limits, and why real machines and processes have unavoidable losses.
Entropy and Disorder (Qualitative View)
On a microscopic level, entropy is related to the number of microscopic arrangements (microstates) compatible with the macroscopic state of a system.
Qualitative trends important for chemistry:
- Gases have higher entropy than liquids, which have higher entropy than solids (at the same temperature and composition).
- A substance at higher temperature generally has higher entropy than the same substance at lower temperature.
- Mixing different substances (e.g. ideal gases, solutions) increases entropy.
- More complex molecules can have higher entropy due to more ways to distribute energy among their internal motions.
A common, but simplified, interpretation is that entropy measures “disorder” or “randomness”. For many chemical applications it is more accurate to view entropy as a measure of the spread of energy and matter over available states.
The Second Law and Equilibrium
The second law not only describes the direction of processes but also the condition for equilibrium:
- Spontaneous processes in an isolated system occur in the direction of increasing entropy.
- Equilibrium in an isolated system is reached when entropy is maximal and no further spontaneous change is possible.
For systems that can exchange energy and/or matter with their surroundings (non-isolated systems), appropriate thermodynamic potentials (like Helmholtz and Gibbs free energy) are used. Their connection to entropy and the second law is treated elsewhere; here, the key idea is:
- The second law determines both the direction of approach to equilibrium and the condition at equilibrium.
In chemical systems, this perspective underpins the later relation between equilibrium, entropy changes, and Gibbs free energy.
Summary of Key Points
- The second law introduces a preferred direction for processes, not contained in the first law.
- It can be expressed in several equivalent ways (Kelvin–Planck, Clausius, entropy formulation).
- Entropy $S$ is a state function defined (for reversible changes) by $dS = \delta Q_\mathrm{rev}/T$.
- For any real process: $\Delta S_\mathrm{universe} \ge 0$.
- Reversible processes are ideal limits with no net entropy production.
- Irreversibility (e.g. spontaneous heat flow, mixing, expansion, chemical reaction) is always accompanied by positive entropy production.
- Entropy tends to increase in spontaneous processes; equilibrium corresponds to an extremum (maximum in isolated systems) of entropy.