The First Law: Internal Energy, Heat, and Work

The first law is the statement that energy is conserved once heat is counted as a form of energy transfer. Before the mechanical equivalent of heat was measured, mechanical energy and thermal energy were separate ledgers; the first law merges them. For a thermodynamic system the merged ledger has a single balance, the internal energy, and two channels through which it changes, heat and work. The content of the law is that these two path-dependent channels always sum to a path-independent change.

Internal energy as a state function

The internal energy $U$ of a system is the total energy of its microscopic constituents in the frame where the system is at rest: kinetic energy of the molecules plus the potential energy of their interactions, excluding the bulk kinetic and potential energy of the system as a whole. For an equilibrium state, $U$ is fixed by the state variables, so it is a state function with an exact differential $\mathbf{d}U$. Around any cycle,

This is an assertion, justified microscopically later by identifying $U$ with the ensemble average of the Hamiltonian, and justified thermodynamically by the first law itself: the existence of an energy function whose change equals the total energy input is the empirical claim.

Work

Work is energy transferred across the system boundary through a macroscopic force acting over a displacement. For a fluid in a cylinder of cross-section $A$ whose piston advances a distance $\mathbf{d}x$ under external pressure $P_{\mathrm{ext}}$, the work done on the system is $\delta W=F\mathbf{d}x=P_{\mathrm{ext}}A\mathbf{d}x=-P_{\mathrm{ext}}\mathbf{d}V$, since advancing the piston decreases the volume, $\mathbf{d}V=-A\mathbf{d}x$. For a quasi-static process the external pressure matches the system pressure up to an infinitesimal, $P_{\mathrm{ext}}=P$, and

The sign convention taken here counts work done on the system as positive.¹ Compression ($\mathbf{d}V<0$) then delivers positive work to the system, expansion ($\mathbf{d}V>0$) extracts it. The integral is the negative of the signed area under the process curve on the $P$–$V$ plane, so it depends on the entire path, not just the endpoints: different curves joining the same two states enclose different areas and transfer different work.

Work has other forms with the same structure: magnetic work $B\mathbf{d}M$, surface work $\gamma \mathbf{d}A$, elastic work, chemical work $\mu \mathbf{d}N$. Each is an intensive force times the differential of its conjugate extensive displacement. The fluid case $-P\mathbf{d}V$ is the running example, and the generalization is carried into the thermodynamic-potential lesson.

Heat and the first law

Energy can also cross the boundary without any macroscopic displacement, driven instead by a temperature difference through a diathermal wall. That transfer is heat, $\delta Q$. Like work, heat is a process quantity: the amount depends on the path, and $\delta Q$ is inexact. The first law asserts that although $\delta Q$ and $\delta W$ are each path-dependent, their sum is the exact differential of the internal energy.

The law does two things at once. It defines heat, as the difference $\mathbf{d}U-\delta W$ between the energy change and the identifiable work — the residual channel. And it constrains the system, because $\mathbf{d}U$ is exact while neither term on the right is: the inexactness of $\delta Q$ and $\delta W$ must cancel. For the quasi-static fluid,

Heat capacities and enthalpy

The heat capacity measures the heat required to raise the temperature, $C=\delta Q/\mathbf{d}T$. Because $\delta Q$ is path-dependent, the value depends on what is held fixed. Two constraints matter.

At constant volume, no work is done, so $\delta Q=\mathbf{d}U$ and

At constant pressure, the system does expansion work as it warms, so part of the heat goes to work and $C_P>C_V$. Writing $\delta Q=\mathbf{d}U+P\mathbf{d}V$ at fixed $P$ suggests grouping $U+PV$ into a single function.

Enthalpy is the natural energy for constant-pressure processes, which describes most chemistry and much of everyday physics, where the atmosphere fixes $P$. It is the first of the thermodynamic potentials, obtained from $U$ by a Legendre transform swapping the conjugate pair $(P,V)$; the systematic construction is the fourth lesson.

For an ideal gas the two capacities differ by a fixed amount. Since $U$ depends on $T$ alone (shown next) and $PV=N{k}_{B}T$,

the Mayer relation for an ideal gas. Their ratio, the adiabatic index$\gamma \equiv C_P/C_V$, governs adiabatic processes below.

Internal energy of the ideal gas

That $U$ depends on temperature alone for an ideal gas is an experimental result — Joule's free-expansion experiment, in which a gas expanding into vacuum through an adiabatic, rigid boundary does no work and exchanges no heat, so $\Delta U=0$, and its temperature is found not to change despite the volume change. Hence $(\partial U/\partial V{)}_T=0$ and

For a monatomic ideal gas the equipartition theorem (derived in a later module) gives $U=\frac{3}{2}N{k}_{B}T$, so $C_V=\frac{3}{2}N{k}_B$, $C_P=\frac{5}{2}N{k}_B$, and $\gamma =5/3$. A diatomic gas at room temperature adds two rotational degrees of freedom, $C_V=\frac{5}{2}N{k}_B$ and $\gamma =7/5$.

Isothermal and adiabatic processes

Two quasi-static processes of the ideal gas recur throughout the subject.

An isothermal process holds $T$ fixed, so $\mathbf{d}U=0$ and all the heat becomes work. Along an isotherm $P=N{k}_{B}T/V$, and

An isotherm is a hyperbola $PV=\text{const}$ on the $P$–$V$ plane.

An adiabatic process exchanges no heat, $\delta Q=0$, so $\mathbf{d}U=\delta W$ and the gas does work entirely at the expense of its internal energy. With $\mathbf{d}U=C_V\mathbf{d}T$ and $\delta W=-P\mathbf{d}V=-N{k}_{B}T\mathbf{d}V/V$,

using $N{k}_B=C_P-C_V$ and $\gamma =C_P/C_V$. Integrating gives $T{V}^{\gamma -1}=\text{const}$, and substituting $T\propto PV$ yields the adiabat

Because $\gamma >1$, the adiabat is steeper than the isotherm at every point: an adiabatic expansion cools the gas, dropping its pressure faster than the isothermal hyperbola would. The two curves cross transversally, and the wedge between them is the working region of the Carnot cycle in the next lesson.

The path dependence of heat and work, and the path independence of their sum, is sharpest in a direct comparison of two routes between the same endpoints.

The heat and work bookkeeping of any process reduces to the same accounting: heat in, work in, and the stored change in internal energy that balances them.

Summary

• Internal energy $U$ is a state function with exact $\mathbf{d}U$; heat $\delta Q$ and work $\delta W$ are path-dependent process quantities.
• The first law $\mathbf{d}U=\delta Q+\delta W$ conserves energy across both channels; the inexactness of the two terms cancels so their sum is exact. Quasi-static fluid work is $\delta W=-P\mathbf{d}V$, the negative signed area under the process curve.
• $C_V=(\partial U/\partial T{)}_V$ and $C_P=(\partial H/\partial T{)}_P$ with $H=U+PV$; for an ideal gas $C_P-C_V=N{k}_B$ and $\gamma =C_P/C_V$.
• An ideal gas has $U=U(T)$. Its isotherm is $PV=\text{const}$ with $Q=N{k}_{B}T\ln (V_b/V_a)$; its adiabat is $P{V}^{\gamma }=\text{const}$, steeper than the isotherm, and adiabatic expansion cools the gas.