First vs Second Law of Thermodynamics: Difference

One law does the accounting, the other picks the direction

The first law is a balance sheet. Essential knowledge 9.4.B.1 calls it a restatement of conservation of energy that accounts for energy transferred into or out of a system by work, heating, or cooling. For a closed system it is written

and it tells you how much internal energy changed. It says nothing about whether the process you just balanced can actually happen.

The second law is a direction sign. Essential knowledge 9.6.A.1 states that the total entropy of an isolated system can never decrease, and is constant only when all processes the system undergoes are reversible. That single restriction is what stops a lukewarm cup of coffee from separating into a scalding half and a freezing half while its total energy stays exactly the same.

Every useful thing on this page comes out of that asymmetry. The first law is an equation and both sides of it run happily backwards: play any energy-conserving film in reverse and the joules still add up. The second law is an inequality, and an inequality has a preferred direction. It is the only statement in AP Physics 2 that distinguishes the future from the past.

So the two are not competing accounts of the same thing, and they are not a stronger and a weaker version of one idea. They answer different questions. How much is the first law. Which way is the second.

Side by side

The last two rows are the pair that gets misquoted. Isolated and closed are not synonyms in this course. An isolated system exchanges nothing, so its total energy is constant and its entropy cannot fall. A closed system exchanges energy but not matter, so it has a first law with real $Q$ and $W$ terms in it, and 9.6.A.3.ii says its entropy is free to decrease because energy can be transferred into or out of it. A freezer lowers the entropy of the water in the tray. It is not a counterexample to anything, because the tray is not isolated.

The case that separates them: two blocks and 4500 joules

Put two identical metal blocks in contact inside a perfect insulator, one at $20 \ ^\circ\mathrm{C}$ and one at $80 \ ^\circ\mathrm{C}$. Each has mass $0.50 \ \mathrm{kg}$ and specific heat $900 \ \mathrm{J/(kg \cdot K)}$.

Now propose an outcome: the cold block cools to $10 \ ^\circ\mathrm{C}$ and the hot block warms to $90 \ ^\circ\mathrm{C}$.

Run the first law over it. Using $Q = mc\Delta T$, the cold block gives up $(0.50)(900)(10) = 4500 \ \mathrm{J}$ and the hot block takes in exactly $(0.50)(900)(10) = 4500 \ \mathrm{J}$. The pair is isolated, the totals cancel, and the energy of the whole system is unchanged. The first law is satisfied to the joule. There is no accounting error to find, because there is no accounting error.

The process still never happens, and the reason is not energetic. Essential knowledge 9.3.A.3 states that energy is transferred through thermal processes spontaneously from a higher-temperature system to a lower-temperature system, and 9.3.A.3.i explains the mechanism: in collisions between atoms from different systems, energy is most likely to be transferred from higher-energy atoms to lower-energy atoms. Concentrating energy in the hot block runs the wrong way, and 9.6.A.2.i puts the point as a tendency: localized energy will tend to disperse and spread out.

What does happen is the outcome the second law allows. Energy runs from the hot block into the cold one until both sit at $50 \ ^\circ\mathrm{C}$, which is thermal equilibrium by 9.3.A.4. That transfer is $(0.50)(900)(30) = 13500 \ \mathrm{J}$ in each direction of the ledger, and the first law is satisfied by this outcome exactly as well as it was by the forbidden one. Both outcomes pass the first law. Only one passes the second. That is the whole page in two sentences.

The first law in AP form, and why every sign follows from one decision

The AP Physics 2 equation sheet prints two lines in the Thermal Physics box that have to be read together:

and its own symbol key settles the convention: $Q$ is energy transferred to a system by heating, and $W$ is work done on a system. Essential knowledge 9.4.B.1.iii matches it, defining $W = -P\Delta V$ as the work done on a system by a constant or average external pressure that changes the volume of that system.

That minus sign is not decoration. It exists to force the sign of $W$ to agree with the phrase "on the system":

Plenty of textbooks write the first law as $\Delta U = Q - W$ with $W$ meaning work done by the gas. That version is self-consistent and returns identical physics. Mixing the two inside one problem is what produces an internal energy change with the wrong sign and no warning, because the arithmetic never complains. Use the sheet's version, and translate any other source before you substitute.

Reading work off a diagram, handling multi-step paths and closing a cycle belong to the PV diagrams guide. What matters here is that none of that machinery has any opinion about direction. You can compute $Q$, $W$ and $\Delta U$ for a process that the second law rules out, and every number will be correct.

How far AP Physics 2 takes the second law

Further than most students expect on the concepts, and not one step into the algebra. The boundary statement under Topic 9.6 is one line:

> Only qualitative treatment of the second law of thermodynamics is within the scope of AP Physics 2.

Render the Table of Information appendix and the Thermal Physics box confirms it. The box prints $P = F_\perp / A$, $K_{avg}$, the conduction rate, the ideal gas law, $U = \frac{3}{2}nRT$, $W = -P\Delta V$, $\Delta U = Q + W$ and $Q = mc\Delta T$. There is no $S$ anywhere on the sheet, no $\Delta S = Q/T$, and no efficiency formula. A second-law answer is written in sentences, and it earns its points by naming an essential knowledge statement rather than by producing a number.

What you are expected to be able to say:

• Entropy is qualitative here. 9.6.A.2 describes it as the tendency of energy to spread, or the unavailability of some of the system's energy to do work. The CED never uses the word "disorder" and never mentions microstates, so an answer built on tidiness has nothing to stand on.
• Entropy is a state function (9.6.A.2.ii), depending only on the current state or configuration of a system and not on how the system reached that state. Internal energy is a state function too. $Q$ and $W$ are not, which is why two paths between the same endpoints can carry different amounts of each and still land on the same $\Delta U$.
• Maximum entropy occurs at thermodynamic equilibrium (9.6.A.2.iii), and isolated systems spontaneously move toward it (9.6.A.3.i).
• The equality case is reversibility. 9.6.A.1 says entropy is constant only when all processes the system undergoes are reversible. Quote the statement without that clause and you have turned a careful law into a false one.

Heat engines appear in this unit only as an essential question in the unit opener, and the course sets no efficiency to calculate. If a problem seems to want a Carnot efficiency, it is not an AP Physics 2 problem.

When it costs a mark

Answering a direction question with an energy argument. Asked why energy will not flow from the cold block to the hot one, a first-law answer says the totals still balance, which is true and worth nothing. The point is in 9.3.A.3 or in the entropy statement, and a response that only conserves energy has answered a question nobody asked.

"Entropy always increases." Two words too strong in two places. The law is about the total entropy of an isolated system, and 9.6.A.3.ii is explicit that a closed system's entropy can decrease. A refrigerator is the standard trap, and the correct move is to widen the system until it is isolated.

Dropping the reversibility clause. "Entropy never decreases" is the half of 9.6.A.1 that people remember. "And is constant only when all processes the system undergoes are reversible" is the half that makes it precise.

Sign slips in the first law. Substituting a positive $W$ for an expansion is the classic. Expansion means $\Delta V > 0$, so $W = -P\Delta V$ is negative on the AP convention, and $\Delta U$ comes out too large by twice the work if you get it backwards.

Treating $\Delta U = 0$ as $Q = 0$. In an isothermal process the internal energy of an ideal gas does not change, so $Q = -W$: energy still crosses the boundary in both forms, and they cancel. Zero net change is not zero traffic. The isothermal vs adiabatic comparison separates that pair.

Writing an entropy equation. There is nothing to substitute into. A quantitative entropy calculation on an AP Physics 2 response is not wrong so much as out of scope, and it usually replaces the qualitative justification that was actually being scored.

Where they agree, and why that hides the asymmetry

For a whole unit's worth of problems the two laws never disagree, and that is exactly why the difference goes unnoticed until it is examined.

Every process you are asked to compute already obeys both. Problems hand you a compression, an isobaric expansion, a cycle. All of them are physically possible, so the second law is silently satisfied before you pick up a pencil and the first law does all the visible work. The second law only shows its teeth on a process someone has proposed and you have to reject.

In the reversible limit they meet. 9.6.A.1's equality case is the idealization where total entropy holds still. A reversible process runs equally well in both directions, so the second law stops distinguishing them, and the first law is all that is left. No real process reaches that limit, which is why the equality case is a boundary rather than a description.

Both are silent about the same thing: rate. Neither law tells you how long any of this takes. That belongs to the conduction relation in Topic 9.5, $\frac{Q}{\Delta t} = \frac{kA\Delta T}{L}$, and it is a separate question from whether the transfer happens at all.

One last coincidence worth naming. For an isolated system the first law says the total energy is constant and the second says the total entropy is nondecreasing. Both are statements about an isolated system, both use the word "total", and the sentences look parallel. They are not: one is a conserved quantity and the other is a quantity with a preferred direction. Conservation and direction are different properties, and a system can hold one fixed while the other climbs.

Where this sits on the AP exam

Both laws live in Unit 9, Thermodynamics, which carries 15 to 18 percent of the multiple-choice section over a suggested 10 to 16 class periods. They are separate topics and the split is the one this page describes.

Topic 9.4, The First Law of Thermodynamics, builds internal energy first (9.4.A.1: the sum of the kinetic energy of the objects making up the system and the potential energy of the configuration of those objects), notes that an ideal gas has no internal potential energy because its atoms do not interact via conservative forces (9.4.A.1.i), and then states the law. Its suggested skills are 1.C, 2.A, 2.C and 3.C, so sketching a graph and comparing two scenarios are as likely as a calculation.

Topic 9.6, Entropy and the Second Law of Thermodynamics, carries skills 1.A, 2.C, 3.B and 3.C. Two of those four are argument skills, which is what a qualitative-only topic looks like on a skills list.

Direction also appears one topic earlier than most people look. 9.3.A.3, in Thermal Energy Transfer and Equilibrium, already fixes the spontaneous direction of energy transfer before entropy has been named. A question about why the cold object warms up can be answered from Topic 9.3 alone.

For the mechanics of the first law on a diagram, see the PV diagrams guide. For the two ways energy crosses a boundary, see heat vs work. For the difference between energy in transit and energy already inside a system, compare the glossary entries for heat and internal energy.