2.1 First law of thermodynamics

Fundamental concepts

The first law of thermodynamics is, at its core, a statement about energy conservation. It tells you that the total energy of an isolated system stays constant: energy can change forms, but it can't appear from nothing or vanish into nothing. In statistical mechanics, this law emerges from the collective behavior of enormous numbers of particles, bridging the gap between what individual molecules do and what you can measure with a thermometer or pressure gauge.

Energy conservation principle

Energy conservation means the total energy of a closed system doesn't change unless energy crosses the system boundary as heat or work. For an isolated system (no energy or matter crossing the boundary), this is expressed as:

$\Delta E_{total} = 0$

This principle underpins every thermodynamic process you'll encounter, from heat engines to chemical reactions to refrigeration cycles.

System and surroundings

A system is whatever specific part of the universe you're analyzing (a gas in a piston, a beaker of reacting chemicals). The surroundings are everything else. Getting the boundary right matters, because the first law tracks energy crossing that boundary.

Three types of systems:

• Open systems allow transfer of both energy and matter across the boundary
• Closed systems permit energy transfer but not matter
• Isolated systems prohibit both energy and matter exchange

Choosing the wrong system type is a common source of errors when applying the first law. Before solving any problem, clearly define what's inside your boundary and what's outside.

Thermodynamic equilibrium

A system is in thermodynamic equilibrium when its macroscopic properties stop changing over time. This requires three conditions to be met simultaneously:

• Thermal equilibrium: uniform temperature throughout the system
• Mechanical equilibrium: uniform pressure (no net forces causing bulk motion)
• Chemical equilibrium: balanced chemical potentials for all components

Equilibrium is what allows you to assign definite values to state variables like $T$, $P$, and $V$. Without it, those quantities aren't well-defined for the system as a whole.

Internal energy

Definition and properties

Internal energy ($U$) is the total energy contained within a thermodynamic system. It includes the kinetic energy of molecular motion (translation, rotation, vibration) and the potential energy from intermolecular forces.

A few key points:

• $U$ depends on the system's temperature, volume, and composition
• You can never measure $U$ directly; you can only measure changes in internal energy ($\Delta U$)
• For an ideal gas with no intermolecular forces, $U$ depends on temperature alone

Extensive vs. intensive variables

Internal energy is an extensive variable, meaning it scales with the size of the system. Double the amount of gas, and you double $U$.

• Extensive properties ($U$, $V$, $S$, $n$) scale linearly with the amount of substance
• Intensive properties ($T$, $P$, $\rho$) stay the same if you divide the system in half

You can always convert an extensive property to an intensive one by dividing by the amount of substance (e.g., molar internal energy $u = U/n$).

State function characteristics

Internal energy is a state function: its value depends only on the current state of the system (specified by variables like $T$, $V$, $n$), not on how the system got there.

This is powerful. If you know the initial and final states, you can calculate $\Delta U$ without knowing anything about the intermediate steps. You can pick whichever path is easiest to compute, and the answer will be the same. Heat and work, by contrast, are not state functions; they depend on the specific path taken.

Work in thermodynamics

Work is energy transfer between a system and its surroundings through organized, macroscopic motion (like a piston moving). It's distinct from heat, which is energy transfer driven by a temperature difference.

Mechanical work

In classical mechanics, work is force applied over a distance:

$W = \int \vec{F} \cdot d\vec{x}$

Sign convention (used throughout this guide): work done by the system on the surroundings is positive. Work done on the system is negative. Some textbooks use the opposite convention, so always check.

Pressure-volume work

The most common form of work in thermodynamics is PV work, which occurs when a system expands or compresses against an external pressure:

$W = \int P_{ext} \, dV$

Note: with the sign convention where $\Delta U = Q - W$, expansion ($dV > 0$) against positive external pressure gives positive work done by the system. On a P-V diagram, the work equals the area under the curve for the process.

For a reversible process, $P_{ext}$ equals the system pressure $P$ at every point, so you can use the system's equation of state to evaluate the integral.

Other forms of work

Thermodynamic work isn't limited to PV work. Other forms include:

• Electrical work: movement of charges through a potential difference
• Magnetic work: changes in magnetization under an applied field
• Surface tension work: expanding or contracting a fluid interface

All of these can be written in a generalized form: $đW = Y \, dX$, where $Y$ is a generalized force and $X$ is a generalized displacement.

Heat transfer

Heat ($Q$) is energy transfer driven by a temperature difference between a system and its surroundings. At the microscopic level, it corresponds to the disordered transfer of energy through molecular collisions and radiation.

Heat as energy transfer

Heat flows spontaneously from regions of higher temperature to regions of lower temperature. It's measured in joules (J) and is a process quantity, not a state function. You can't say a system "contains" a certain amount of heat; you can only say a certain amount of heat was transferred during a process.

Conduction, convection, and radiation

The three mechanisms of heat transfer:

• Conduction: direct energy transfer through particle collisions, dominant in solids. Governed by Fourier's law: $q = -k \nabla T$.
• Convection: energy transfer through bulk fluid motion. Can be natural (driven by buoyancy) or forced (driven by fans or pumps).
• Radiation: energy transfer via electromagnetic waves. Governed by the Stefan-Boltzmann law for blackbodies. Dominates at high temperatures or across vacuum.

Sign convention for heat

• $Q > 0$: heat absorbed by the system
• $Q < 0$: heat released by the system

This is consistent with the work convention used in $\Delta U = Q - W$, so that both positive $Q$ and negative $W$ (work done on the system) increase internal energy.

Mathematical formulation

First law equation

For a closed system, the first law is:

$\Delta U = Q - W$

where:

• $\Delta U$ = change in internal energy (state function)
• $Q$ = heat transferred to the system
• $W$ = work done by the system

This applies to both reversible and irreversible processes. The individual values of $Q$ and $W$ depend on the path, but their difference $Q - W$ always equals $\Delta U$, which depends only on the endpoints.

Differential form

For infinitesimal changes:

$dU = đQ - đW$

The "đ" notation (sometimes written $\delta$) on $Q$ and $W$ indicates inexact differentials, meaning these quantities are path-dependent. By contrast, $dU$ is an exact differential. This distinction is fundamental: you can integrate $dU$ between two states and get a unique answer, but integrating $đQ$ or $đW$ requires knowing the specific process path.

Integral form

Over a finite process from state $i$ to state $f$:

$\Delta U = \int_{i}^{f} đQ - \int_{i}^{f} đW$

For a cyclic process (system returns to its initial state), $\Delta U = 0$, which means:

$Q_{net} = W_{net}$

The net heat absorbed over a complete cycle equals the net work done by the system. This result is the foundation for analyzing heat engines and refrigerators.

Processes and applications

Isothermal processes

In an isothermal process, temperature stays constant ($\Delta T = 0$). For an ideal gas, internal energy depends only on temperature, so $\Delta U = 0$ and therefore $Q = W$.

Key relations for an ideal gas:

• $PV = \text{constant}$ (Boyle's law)
• Work done: $W = nRT \ln\left(\frac{V_f}{V_i}\right)$

The system must exchange heat with its surroundings continuously to keep the temperature fixed while doing work.

Adiabatic processes

In an adiabatic process, no heat crosses the system boundary ($Q = 0$), so $\Delta U = -W$. All energy changes come from work alone.

For an ideal gas undergoing a reversible adiabatic process:

• $PV^{\gamma} = \text{constant}$
• $TV^{\gamma - 1} = \text{constant}$

Here $\gamma = C_p / C_v$ is the heat capacity ratio. Because no heat enters or leaves, the temperature changes as work is done: compression heats the gas, expansion cools it. Diesel engine compression and the cooling of rising air parcels in the atmosphere are classic examples.

Isobaric and isochoric processes

Isobaric (constant pressure):

• Work: $W = P \Delta V$
• Heat transfer: $Q_p = \Delta H$ (enthalpy change)
• Common in open-air chemical reactions and constant-pressure calorimetry

Isochoric (constant volume):

• No PV work: $W = 0$ (volume doesn't change)
• Heat transfer: $Q_v = \Delta U$ (all heat goes directly into internal energy)
• Occurs in rigid sealed containers, as in bomb calorimetry

Enthalpy

Definition and significance

Enthalpy is defined as:

$H = U + PV$

Like internal energy, enthalpy is a state function. Its real usefulness shows up at constant pressure: the change in enthalpy equals the heat transferred.

$\Delta H = Q_p$

This makes enthalpy the natural quantity for describing heat of reaction, heat of formation, and latent heats of phase transitions, since most laboratory and real-world processes happen at roughly constant (atmospheric) pressure.

Relationship to internal energy

The connection between enthalpy and internal energy changes is:

$\Delta H = \Delta U + \Delta(PV)$

At constant pressure, this simplifies to $\Delta H = \Delta U + P\Delta V$. For an ideal gas, since $PV = nRT$:

$H = U + nRT$

The difference between $\Delta H$ and $\Delta U$ is the PV work term. For reactions involving only solids and liquids (small volume changes), $\Delta H \approx \Delta U$. For reactions involving gases, the difference can be significant.

Constant pressure processes

At constant pressure, measuring heat flow directly gives you $\Delta H$. This is why chemists report standard enthalpies of formation ($\Delta H_f^\circ$) and reaction ($\Delta H_{rxn}^\circ$) rather than internal energy changes. Constant-pressure calorimeters (like coffee-cup calorimeters) measure $\Delta H$ directly.

Specific heats

Specific heats (heat capacities) tell you how much energy is needed to raise a substance's temperature by one degree. In statistical mechanics, they connect directly to the number of microscopic degrees of freedom available to store energy.

Heat capacity at constant volume

$C_v = \left(\frac{\partial U}{\partial T}\right)_V$

This measures how internal energy changes with temperature when volume is held fixed. No PV work is involved, so all added energy goes into raising the temperature.

For ideal gases, the equipartition theorem predicts:

• Monatomic (3 translational DOF): $C_v = \frac{3}{2}R \approx 12.5 \text{ J/(mol·K)}$
• Diatomic at room temperature (3 translational + 2 rotational DOF): $C_v = \frac{5}{2}R \approx 20.8 \text{ J/(mol·K)}$

Heat capacity at constant pressure

$C_p = \left(\frac{\partial H}{\partial T}\right)_P$

At constant pressure, some of the added heat goes into expanding the system against external pressure, so you need more heat per degree of temperature rise. This is why $C_p > C_v$ for virtually all substances.

For any ideal gas:

$C_p = C_v + R$

This is known as Mayer's relation.

Relationship between specific heats

The ratio $\gamma = C_p / C_v$ appears throughout thermodynamics, especially in adiabatic process equations.

These values come from the equipartition theorem in statistical mechanics, which assigns $\frac{1}{2}k_BT$ of energy per degree of freedom per molecule. At very high temperatures, vibrational modes activate and $\gamma$ decreases further.

Cyclic processes

Cyclic processes bring a system back to its starting state, so $\Delta U = 0$ and $Q_{net} = W_{net}$. They form the theoretical basis for heat engines (converting heat to work) and refrigerators (using work to move heat against a temperature gradient).

Carnot cycle

The Carnot cycle is the most efficient possible cycle operating between two thermal reservoirs at temperatures $T_H$ (hot) and $T_C$ (cold). It consists of four reversible steps:

1. Isothermal expansion at $T_H$ (absorb heat $Q_H$)
2. Adiabatic expansion (cool from $T_H$ to $T_C$)
3. Isothermal compression at $T_C$ (reject heat $Q_C$)
4. Adiabatic compression (warm from $T_C$ back to $T_H$)

Maximum efficiency:

$\eta_{Carnot} = 1 - \frac{T_C}{T_H}$

where temperatures must be in Kelvin. No real engine can exceed this efficiency. For example, a Carnot engine operating between 600 K and 300 K has a maximum efficiency of 50%.

Otto cycle

The Otto cycle models spark-ignition (gasoline) engines. Its four idealized steps are:

1. Isentropic (adiabatic, reversible) compression
2. Isochoric (constant volume) heat addition
3. Isentropic expansion
4. Isochoric heat rejection

Thermal efficiency:

$\eta_{Otto} = 1 - \frac{1}{r^{\gamma - 1}}$

where $r$ is the compression ratio ($V_{max}/V_{min}$). Higher compression ratios give better efficiency, but practical limits like engine knock constrain $r$ to around 8-12 for gasoline engines.

Diesel cycle

The Diesel cycle models compression-ignition engines. It differs from the Otto cycle in that heat addition occurs at constant pressure (isobaric) rather than constant volume. Diesel engines use higher compression ratios (14-25) because they compress air alone before injecting fuel.

Diesel cycle efficiency depends on both the compression ratio and the cutoff ratio (ratio of volume after to volume before combustion). For the same compression ratio, the Otto cycle is more efficient, but diesel engines achieve higher compression ratios in practice, making them more efficient overall.

First law limitations

Reversible vs. irreversible processes

A reversible process proceeds through a continuous sequence of equilibrium states and can be reversed without any net change in the system or surroundings. An irreversible process involves departures from equilibrium (friction, rapid expansion, mixing, heat flow across a finite temperature difference) and always produces entropy.

The first law applies equally to both. It doesn't distinguish between them, which is precisely its limitation.

Direction of spontaneous processes

The first law tells you that energy is conserved, but it says nothing about which direction a process will go. A hot cup of coffee cooling to room temperature and a room-temperature cup spontaneously heating up both conserve energy. The first law permits both, yet only one actually happens.

Similarly, the first law can't explain why gases expand to fill their container, why mixing is spontaneous, or why you can't unscramble an egg. Something beyond energy conservation is needed.

Need for the second law

The second law of thermodynamics fills this gap by introducing entropy ($S$), a state function that quantifies the degree of microscopic disorder. The second law states that the total entropy of an isolated system never decreases:

$\Delta S_{total} \geq 0$

This provides a criterion for spontaneity, establishes fundamental efficiency limits (like the Carnot bound), and connects the statistical mechanics picture of microstates to macroscopic irreversibility. The first law tells you the energy budget; the second law tells you which transactions are actually allowed.