The first law: internal energy, work and heat

Energy is conserved, so only heat and work can change a system

Intuition

Your bank balance only changes if money crosses the boundary: a deposit or a withdrawal. The first law says the same about energy. A system’s internal energy changes only when heat or work moves energy across its boundary. It never appears from nowhere.

Lecture 13 gave a microscopic picture, statistical mechanics. Thermodynamics looks at the macroscopic scale, where a system is described by bulk properties such as volume, pressure and temperature. The lecture’s plan is to revisit the laws of classical thermodynamics and show that they can be derived from the microscopic picture.

The four laws, as the slide states them:

0. If two systems are each in thermal equilibrium with a third, they are in thermal equilibrium with each other.
1. Energy is conserved.
2. The entropy of an isolated system always increases.
3. The entropy of a system approaches a constant value as the temperature goes to zero.

The zeroth law defines thermal equilibrium. We saw that one parameter, the temperature, fixes the equilibrium distribution, so systems at the same temperature are in equilibrium with each other.

The first law comes in two versions. Version 1: energy is conserved. Version 2: the internal energy of an isolated system is constant. A boundary decides what can cross it. An open system exchanges matter and energy, a closed system exchanges energy only, and an isolated system exchanges neither. For the universe as system plus surroundings:

$\Delta U_{\text{total}} = \Delta U_{\text{system}} + \Delta U_{\text{surroundings}} = 0 \quad\Rightarrow\quad \Delta U_{\text{system}} = -\Delta U_{\text{surroundings}}$

What one side gains, the other loses. Microscopically, the internal energy is the total energy of the occupied microstates:

$U = E = \sum_i n_i \varepsilon_i$

The first law: change in U equals heat in plus work done on the system

How can $U$ change? Only through heat $Q$ entering and work $W$ done on the system:

$\Delta U = Q + W$

The sign convention matters. In this lecture $Q$ is positive when heat enters the system, and $W$ is positive when work is done on the system. The slide notes that some books, and Wikipedia, write $\Delta U = Q - W$, where $W$ is the work done by the system. The two are the same law with a different sign for work. Follow the lecture form in this unit.

Heat and work are also transient. They describe energy in motion, not energy sitting inside. Picture water running into a bath: the bath holds water, not “flow”. A system holds internal energy, not heat or work.

Pitfall

Check which convention a question uses before you put a sign on $W$. With the lecture convention ($\Delta U = Q + W$), a gas that expands and pushes its surroundings away has negative $W$, because the system loses energy that way.

Work is a force acting through a displacement

Work is done when a macroscopic force makes something move:

$dW = \mathbf{F}\cdot d\mathbf{x}$

The standard case is a gas pushing a piston. The slide’s piston has a mass on top and mechanical stops, drawn at the initial and final state. The work is

$dW = -p_{\text{ext}}\,dV, \qquad W = -\int_{V_i}^{V_f} p_{\text{ext}}\,dV$

Why the minus sign? If the gas expands, $dV$ is positive, but the system hands energy to the surroundings, so its energy falls and $dW$ is negative. If you squeeze the gas, $dV$ is negative and $dW$ is positive: you put energy in.

More generally, work is a generalised force acting through a generalised displacement. The slide lists:

Type of work	dw	Quantities
Expansion	$-p_{\text{ex}}\,dV$	external pressure, change in volume
Surface expansion	$\gamma\,d\sigma$	surface tension, change in area
Extension	$f\,dl$	tension, change in length
Electrical	$\phi\,dQ$	electric potential, change in charge

Heat is any energy flow into or out of a system that is not work. It is the energy that crosses the boundary because of a temperature difference between the system and its surroundings. So work is organised movement along a force, and heat is everything else.

At the molecular level, work shifts the levels and heat moves the molecules

Intuition

Picture molecules as people standing on shelves in a cupboard. You can raise or lower a shelf with the people on it. Or you can lift people from a low shelf to a high shelf. Both change the total energy, but they are different jobs.

Start from $U = \sum_i n_i \varepsilon_i$ and vary both the occupations $n_i$ and the level energies $\varepsilon_i$:

$dU = \sum_i n_i\,d\varepsilon_i + \sum_i \varepsilon_i\,dn_i$

The slide labels the first sum work: the energy levels themselves shift, for example when a gas’s box changes size. The second sum is heat: the levels stay put but the populations $n_i$ change as molecules move between levels. This is the microscopic meaning of the first law.

Exam detail

Lecture 14 summary: the four laws are not fundamental, but can be derived from statistical mechanics. First law: energy is conserved. Energy, heat and work have microscopic interpretations. The sign convention to quote is $\Delta U = Q + W$ with $W$ the work done on the system.

Recall

In dU = sum n_i d(epsilon_i) + sum epsilon_i dn_i, which term is work and which is heat?

The first term, where the levels shift, is work. The second term, where populations change, is heat.