Two Faces of Heat: Thermodynamics and Statistics
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You should be comfortable with basic calculus, specifically partial derivatives, and have a general intuition for what temperature and heat mean from introductory physics.
Why this matters
Thermodynamics and statistical mechanics are two different languages describing the same physical reality.
For centuries, physicists described heat using macroscopic variables like pressure, volume, and temperature, without knowing what atoms were doing. Later, we discovered that these macroscopic laws emerge naturally from the statistical behaviour of vast numbers of microscopic particles. This lecture introduces the "two faces" of thermal physics: the phenomenological approach of thermodynamics, which tells us what is possible, and the statistical approach, which explains why it is so. Understanding this duality is crucial because it allows us to use thermodynamic relations to predict macroscopic behaviour while using statistical mechanics to derive those relations from first principles.
The Macroscopic vs Microscopic View
Imagine a sealed container filled with gas. In the thermodynamic view, you only care about the bulk properties: the pressure exerted on the walls, the volume of the container, and the temperature measured by a thermometer. You do not care about the position or velocity of any single molecule. This is a coarse-grained description. It is incredibly powerful because it requires very little information to predict the system's behaviour under changes in conditions.
In contrast, the statistical view looks at the individual molecules. Each molecule has a specific position and momentum, defining a "microstate." There are an astronomical number of possible microstates for a macroscopic system. The key insight is that while the system fluctuates between these microstates, the macroscopic variables remain stable because the number of microstates corresponding to a given macrostate (like a specific pressure and temperature) is overwhelmingly larger than those corresponding to any other macrostate. This statistical dominance is what we perceive as the "flow" of heat and the direction of time.
The Second Law and Entropy
The most profound statement in thermodynamics is the Second Law. It states that the total entropy of an isolated system can never decrease over time. Entropy, denoted by \( S \), is a state function that measures the disorder or, more precisely, the number of accessible microstates. In thermodynamics, entropy is defined through the reversible heat transfer \( \delta Q_{\text{rev}} \) and temperature \( T \):
This definition is purely phenomenological. It tells us how to calculate changes in entropy for reversible processes. However, it does not explain why entropy increases. The statistical interpretation resolves this. Boltzmann’s famous formula relates entropy to the number of microstates \( \Omega \) corresponding to a macrostate:
Here, \( k_B \) is the Boltzmann constant. For two independent subsystems the numbers of microstates multiply, \( \Omega = \Omega_1 \Omega_2 \), so their entropies add: the logarithm is what makes \( S \) extensive. The Second Law is then a statement of probability: systems evolve towards macrostates with the largest \( \Omega \) because there are simply more ways to be in that state than in any other. This probabilistic view explains why we never see heat flow spontaneously from a cold body to a hot one—it is not forbidden, but it is statistically negligible for macroscopic systems.
Thermodynamic Potentials
While internal energy \( U \) is the most fundamental thermodynamic potential, it is often inconvenient for experiments conducted at constant temperature or pressure. We define other potentials by performing Legendre transforms on \( U \). The most common is the Helmholtz free energy \( F \), defined as:
The differential form of \( F \) is particularly useful for systems at constant temperature and volume:
Similarly, the Gibbs free energy \( G \) is defined as \( G = U + PV - TS \), and its differential is:
These potentials are central to the unit because they determine the direction of spontaneous change under specific constraints. For example, at constant temperature and pressure, a process is spontaneous if \( dG \lt 0 \). The statistical mechanics counterpart to these potentials is the partition function \( Z \). The Helmholtz free energy is directly related to the canonical partition function by \( F = -k_B T \ln Z \). This equation is the bridge between the microscopic world of quantum states and the macroscopic world of thermodynamics.
Maxwell Relations
Because thermodynamic potentials are state functions, their second derivatives must be independent of the order of differentiation. This mathematical property leads to the Maxwell relations, which allow us to relate quantities that are difficult to measure directly (like changes in entropy) to quantities that are easy to measure (like pressure, volume, and temperature). For example, starting from the Helmholtz free energy \( F(T, V) \), we have:
This relation tells us that the change in entropy with respect to volume at constant temperature is equal to the change in pressure with respect to temperature at constant volume. The right-hand side can be measured using a pressure gauge and a thermometer, while the left-hand side involves entropy, which is not directly measurable. Maxwell relations are essential tools for calculating thermodynamic properties from experimental data. They are a direct consequence of the exactness of the thermodynamic differentials and hold true regardless of the microscopic details of the system.
A common misconception is that entropy is a measure of "disorder" in a colloquial sense. While this is a useful heuristic, it is imprecise. Entropy is strictly a measure of the number of microstates compatible with a macrostate. A system can appear "ordered" macroscopically but have high entropy if there are many microscopic configurations that look the same from a distance. Conversely, a "disordered" macrostate might have low entropy if it corresponds to a unique microstate. Always rely on the statistical definition \( S = k_B \ln \Omega \) for rigorous work. C
Worked example
Problem. Consider an ideal monatomic gas in a container of volume \( V = 2.0 \times 10^{-3} \ \mathrm{m^3} \) at a temperature \( T = 300 \ \mathrm{K} \). The gas consists of \( N = 1.0 \times 10^{23} \) molecules. Calculate the pressure \( P \) of the gas and the change in Helmholtz free energy \( \Delta F \) if the gas is isothermally expanded to a volume \( V' = 4.0 \times 10^{-3} \ \mathrm{m^3} \). Use \( R = 8.314 \ \mathrm{J/(mol \cdot K)} \) and \( N_A = 6.022 \times 10^{23} \ \mathrm{mol^{-1}} \).
- First, determine the number of moles \( n \) of the gas: \[ n = \frac{N}{N_A} = \frac{1.0 \times 10^{23}}{6.022 \times 10^{23}} \approx 0.166 \ \mathrm{mol} \]
- Calculate the initial pressure \( P \) using the ideal gas law \( PV = nRT \): \[ P = \frac{nRT}{V} = \frac{0.166 \times 8.314 \times 300}{2.0 \times 10^{-3}} \] \[ P = \frac{413.9}{2.0 \times 10^{-3}} \approx 2.07 \times 10^5 \ \mathrm{Pa} \]
- For an ideal gas, the Helmholtz free energy is given by \( F = U - TS \). For an isothermal process, \( \Delta U = 0 \) (since internal energy depends only on temperature for an ideal gas). Thus, \( \Delta F = -T \Delta S \). The change in entropy for an isothermal expansion of an ideal gas is: \[ \Delta S = nR \ln \left( \frac{V'}{V} \right) \] \[ \Delta S = 0.166 \times 8.314 \times \ln \left( \frac{4.0 \times 10^{-3}}{2.0 \times 10^{-3}} \right) \] \[ \Delta S = 1.38 \times \ln(2) \approx 1.38 \times 0.693 \approx 0.956 \ \mathrm{J/K} \]
- Calculate \( \Delta F \): \[ \Delta F = -T \Delta S = -300 \times 0.956 \approx -286.8 \ \mathrm{J} \]
Units check. Pressure is in Pascals (\( \mathrm{Pa} = \mathrm{N/m^2} \)), which is consistent with \( \mathrm{J/m^3} \). Free energy is in Joules, consistent with \( \mathrm{K} \cdot \mathrm{J/K} \). The negative \( \Delta F \) means that at constant temperature the gas can deliver at most 287 J of work during this expansion (for a reversible isothermal expansion, \( W = -\Delta F \)).
Three quick tests
- What is the physical significance of the Boltzmann constant \( k_B \) in the context of entropy?
Answer
It sets the scale of entropy relative to the number of microstates, linking the microscopic world (microstates) to the macroscopic world (thermodynamic entropy). - Why is the Helmholtz free energy \( F \) preferred over internal energy \( U \) for processes at constant temperature?
Answer
Because \( F \) incorporates the entropy term \( -TS \), allowing it to account for the energy available to do work when the system exchanges heat with a thermal reservoir at temperature \( T \). - What does a Maxwell relation allow you to do?
Answer
It allows you to replace a difficult-to-measure thermodynamic derivative (usually involving entropy) with an equivalent derivative of measurable quantities (like pressure, volume, and temperature).
Problem set for PU-203 →
Each derivation in this unit ends with five graded problems with full solutions.
Lecture notes checked line by line by the editor; every number in the worked example was recomputed.