Energy Accounting and Directionality in Thermodynamics (AP Physics 2, Unit 1)

First Law of Thermodynamics

What the First Law says (and what it’s really about)

The First Law of Thermodynamics is the energy-conservation rule for thermal systems. It tells you how to “keep the books” when energy moves into or out of a system as heat and work, and how that changes the system’s stored energy.

In everyday language: if energy enters a system (as heat) or leaves/enters via mechanical interaction (work), the system’s internal energy changes accordingly. Nothing about energy is created or destroyed—thermodynamics just forces you to be explicit about how energy crossed the boundary.

The standard AP Physics sign convention is:

$\Delta U = Q - W$

• Internal energy $U$: the microscopic energy stored in the system (random translational/rotational/vibrational kinetic energy of particles, plus molecular potential energy). You typically don’t know $U$ itself—what matters is the change $\Delta U$.
• Heat $Q$: energy transferred because of a temperature difference between the system and surroundings.
  • $Q > 0$ means heat flows into the system.
  • $Q < 0$ means heat flows out of the system.
• Work $W$: energy transferred by a force through a distance at the system boundary (often via expansion/compression against an external pressure).
  • In this convention, $W > 0$ means the system does work on the surroundings (the system loses energy by doing work).
  • $W < 0$ means the surroundings do work on the system (energy is added to the system via work).

This sign convention is one of the biggest sources of mistakes—always translate the physical story (“gas expands and pushes a piston”) into the sign (“system did work, so $W > 0$”).

Why it matters

The First Law is the bridge between “thermal” ideas (heat, temperature) and “mechanical” ideas (work, energy conservation). In AP Physics 2, it’s the foundation for:

• Understanding how engines and refrigerators move energy around
• Interpreting $P$–$V$ processes (expansion/compression) using energy ideas
• Recognizing which quantities depend on the path of a process (heat, work) versus only the state (internal energy)

How it works: state functions vs. path-dependent quantities

A key conceptual leap is that internal energy is a state function, while heat and work are not.

• A state function depends only on the current state (for example, pressure/volume/temperature for a gas). If you go from state A to state B, $\Delta U$ is the same no matter how you got there.
• Heat $Q$ and **work** $W$ depend on the _process_ (the path) taken between A and B. Different paths can involve different amounts of heat transfer and work, even though $\Delta U$ ends up the same.

This is why the First Law is so useful: even if you can’t easily compute both $Q$ and $W$, knowing one plus $\Delta U$ gives you the other.

Work in thermodynamic processes (especially gases)

In many AP problems, the “system” is a gas in a piston-cylinder. Mechanical work comes from expansion or compression.

For a quasi-static (slow, well-defined pressure) process, the work done by the gas is the area under the curve on a $P$–$V$ graph:

$W = \int P\,dV$

AP Physics 2 is algebra-based, so you’ll most often use simple cases where $P$ is constant:

$W = P\Delta V$

• If $\Delta V > 0$ (expansion), then $W > 0$: the system does work.
• If $\Delta V < 0$ (compression), then $W < 0$: work is done on the system.

Even if you don’t compute the integral, you should interpret “area under the curve” conceptually: bigger pressure or bigger volume change means more work.

Connecting the First Law to special process descriptions

You’ll often see processes described by what is held constant. The First Law helps you translate those words into energy consequences.

Isochoric (constant volume)

If $\Delta V = 0$, then

$W = 0$

So the First Law becomes:

$\Delta U = Q$

Meaning: any heat added goes entirely into changing internal energy (often raising temperature).

Adiabatic (no heat transfer)

If the system is insulated or the process is so fast that heat can’t flow, then

$Q = 0$

So:

$\Delta U = -W$

Meaning: the only way to change internal energy is through work. For example, compressing a gas quickly can raise its temperature even without adding heat.

Cyclic processes (returns to the same state)

If the system completes a cycle and returns to its original state, then the internal energy returns to its original value:

$\Delta U_{cycle} = 0$

Therefore, over a full cycle:

$Q_{net} = W_{net}$

This is essential for heat engines: they turn net heat input into net work output, but the working substance ends where it started.

Worked examples (First Law)

Example 1: Simple energy accounting with signs

A gas absorbs heat $Q = 600\ \text{J}$ and expands, doing work $W = 200\ \text{J}$ on a piston. Find $\Delta U$.

Start with the First Law:

$\Delta U = Q - W$

Substitute:

$\Delta U = 600\ \text{J} - 200\ \text{J} = 400\ \text{J}$

Interpretation: the gas gained internal energy overall; some incoming heat became work, and the rest increased internal energy.

Example 2: Compression with heat leaving

A gas is compressed so that work done on the gas is $300\ \text{J}$, and during the compression the gas releases heat $Q = -100\ \text{J}$. Find $\Delta U$.

“Work done on the gas” means $W < 0$ in the AP convention. So:

$W = -300\ \text{J}$

Now apply the First Law:

$\Delta U = Q - W = (-100\ \text{J}) - (-300\ \text{J}) = 200\ \text{J}$

Interpretation: even though heat left the system, the compression added more energy by work, so internal energy still increased.

What goes wrong: common conceptual traps

• Confusing heat with internal energy: heat $Q$ is _energy in transit_ due to temperature difference; internal energy $U$ is stored microscopic energy.
• Sign errors on work: if the system expands, it does work on the surroundings, so $W > 0$ and internal energy tends to decrease unless heat flows in.
• Assuming “no temperature change” implies $\Delta U = 0$ in every situation: for an ideal gas, internal energy depends only on temperature, but AP problems won’t always require ideal-gas assumptions. Use what’s given.

Exam Focus

• Typical question patterns:
  • Given two of $Q$, $W$, and $\Delta U$, solve for the third and interpret the signs.
  • Use process keywords (adiabatic, isochoric, cyclic) to simplify the First Law.
  • Connect a $P$–$V$ graph to work (area under the curve) and then to $\Delta U$ via the First Law.
• Common mistakes:
  • Treating “work done on the gas” as $W > 0$ (AP convention uses $\Delta U = Q - W$).
  • Forgetting that in a full cycle $\Delta U_{cycle} = 0$, so $Q_{net} = W_{net}$.
  • Using $W = P\Delta V$ when pressure is clearly not constant (instead, reason with areas or break into segments).

Second Law of Thermodynamics and Entropy

The big idea: energy conservation is not enough

The First Law tells you what’s allowed by energy conservation, but it doesn’t tell you what actually happens spontaneously. For example, energy conservation doesn’t forbid heat from flowing from cold to hot—but in real life, that doesn’t happen on its own.

The Second Law of Thermodynamics introduces directionality: natural processes have a preferred direction, and that direction is connected to the spread/dispersal of energy.

There are two widely used statements of the Second Law (both describe the same underlying rule):

• Kelvin–Planck statement: No device operating in a cycle can take heat from a single thermal reservoir and convert it entirely into work.
• Clausius statement: Heat does not spontaneously flow from a colder object to a hotter object.

These aren’t just “rules to memorize”—they explain why engines must dump waste heat, and why refrigerators require input work.

Heat engines: where the Second Law shows up on exams

A heat engine is a cyclic device that converts some heat input into work output. It operates between:

• a hot reservoir at temperature $T_H$, supplying heat $Q_H$
• a cold reservoir at temperature $T_C$, receiving rejected heat $Q_C$

Over a full cycle, $\Delta U = 0$, so the net work output equals the net heat absorbed:

$W_{out} = Q_H - Q_C$

The thermal efficiency $e$ of an engine is the fraction of input heat that becomes useful work:

$e = \frac{W_{out}}{Q_H}$

Using the energy relation above:

$e = 1 - \frac{Q_C}{Q_H}$

The Second Law implies a crucial limitation: you can’t make $Q_C = 0$ for a cyclic engine. Some energy must be expelled to a colder reservoir; therefore, $e < 1$ for any real engine.

Refrigerators and heat pumps: moving heat “uphill” requires work

A refrigerator is essentially a heat engine run “backwards”: it uses input work to move heat from cold to hot.

• Heat removed from the cold region: $Q_C$
• Heat expelled to the hot environment: $Q_H$
• Work input: $W_{in}$

Energy conservation over a cycle gives:

$Q_H = Q_C + W_{in}$

Because refrigerators are about moving heat, not producing work, we measure performance using a coefficient of performance (COP), not an efficiency.

For a refrigerator:

$K_R = \frac{Q_C}{W_{in}}$

For a heat pump (goal: deliver heat to the warm interior):

$K_{HP} = \frac{Q_H}{W_{in}}$

A common misconception is thinking COP must be less than 1 because “it’s like efficiency.” COP can be greater than 1 because the device isn’t creating energy—it’s transporting heat using work.

The absolute theoretical limit: the Carnot engine

A Carnot engine is an idealized, reversible engine that sets the maximum possible efficiency for any engine operating between the same reservoir temperatures.

Its efficiency depends only on the temperatures (in kelvins):

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

This equation is powerful because it tells you what cannot be beaten, no matter how clever the engineering is. It also encodes a key Second Law message: to get high efficiency, you need a very hot $T_H$ and/or a very cold $T_C$—and neither extreme is always practical.

Entropy: a quantitative way to express the Second Law

To go beyond “things tend to happen one way,” we define entropy $S$, a state function that tracks how energy is spread out and how many microscopic arrangements are consistent with the macroscopic state.

A very useful way to think about entropy in AP Physics 2 is:

• Higher entropy means energy is more dispersed among particles and less available to do organized work.
• Many spontaneous processes (heat flowing hot to cold, gases mixing, friction converting mechanical energy to thermal energy) increase the overall dispersion of energy.

For a reversible transfer of heat at constant temperature $T$, the entropy change is defined as:

$\Delta S = \frac{Q_{rev}}{T}$

Important details:

• $T$ must be in kelvins.
• The formula uses $Q_{rev}$ (reversible heat), which is a theoretical idealization. In practice, you often compute entropy changes by imagining a reversible path between the same initial and final states.

The entropy rule for the universe

The Second Law can be stated compactly using entropy:

$\Delta S_{univ} = \Delta S_{sys} + \Delta S_{surr} \ge 0$

• If $\Delta S_{univ} > 0$, the process is spontaneous (natural direction).
• If $\Delta S_{univ} = 0$, the process is reversible (an ideal limit).
• If $\Delta S_{univ} < 0$, the process cannot occur as stated (it would require external intervention).

This is the deep reason engines have limits: even though energy is conserved, converting dispersed thermal energy entirely into organized work would reduce entropy too much.

Entropy changes in common AP situations

Heat flowing between two reservoirs

Suppose an amount of heat $Q$ flows from a hot reservoir at $T_H$ to a cold reservoir at $T_C$.

• Hot reservoir loses heat: $\Delta S_H = -\frac{Q}{T_H}$
• Cold reservoir gains heat: $\Delta S_C = \frac{Q}{T_C}$

So the total is:

$\Delta S_{univ} = Q\left(\frac{1}{T_C} - \frac{1}{T_H}\right)$

Because $T_C < T_H$, you have $\frac{1}{T_C} > \frac{1}{T_H}$, making $\Delta S_{univ} > 0$. That’s the mathematical statement that heat spontaneously flows hot to cold.

Phase changes at constant temperature

During melting or boiling at the phase-change temperature, the substance absorbs heat without changing temperature. If the phase change is done reversibly at temperature $T$, then:

$\Delta S = \frac{Q}{T}$

Often, $Q$ is given by latent heat:

$Q = mL$

So:

$\Delta S = \frac{mL}{T}$

This is a common AP context because it connects entropy to calorimetry ideas without requiring advanced math.

Worked examples (Second Law, engines, entropy)

Example 1: Engine work and efficiency from heat flows

A heat engine absorbs $Q_H = 1500\ \text{J}$ and exhausts $Q_C = 900\ \text{J}$ each cycle.

Net work output:

$W_{out} = Q_H - Q_C = 1500\ \text{J} - 900\ \text{J} = 600\ \text{J}$

Efficiency:

$e = \frac{W_{out}}{Q_H} = \frac{600\ \text{J}}{1500\ \text{J}} = 0.40$

Interpretation: $40\%$ of the absorbed heat becomes work; the rest must be dumped as waste heat, consistent with the Second Law.

Example 2: Carnot maximum efficiency

An engine operates between $T_H = 600\ \text{K}$ and $T_C = 300\ \text{K}$. What is the maximum possible efficiency?

Use the Carnot limit:

$e_{Carnot} = 1 - \frac{T_C}{T_H} = 1 - \frac{300\ \text{K}}{600\ \text{K}} = 1 - 0.50 = 0.50$

So no engine between these two reservoirs can exceed $50\%$ efficiency.

Example 3: Entropy change for heat transfer between reservoirs

Heat $Q = 1000\ \text{J}$ flows from a reservoir at $T_H = 400\ \text{K}$ to one at $T_C = 300\ \text{K}$.

Entropy change of hot reservoir:

$\Delta S_H = -\frac{Q}{T_H} = -\frac{1000\ \text{J}}{400\ \text{K}} = -2.5\ \text{J/K}$

Entropy change of cold reservoir:

$\Delta S_C = \frac{Q}{T_C} = \frac{1000\ \text{J}}{300\ \text{K}} \approx 3.33\ \text{J/K}$

Total:

$\Delta S_{univ} = \Delta S_H + \Delta S_C \approx (-2.5 + 3.33)\ \text{J/K} \approx 0.83\ \text{J/K}$

Because $\Delta S_{univ} > 0$, the direction (hot to cold) is spontaneous.

What goes wrong: misconceptions about the Second Law and entropy

• “The Second Law says energy is not conserved.” Incorrect. Energy is conserved (First Law). The Second Law restricts which energy transfers and conversions are naturally possible.
• “Heat can never go from cold to hot.” It can, but not spontaneously. Refrigerators do it by using input work $W_{in}$.
• “Entropy means disorder, period.” “Disorder” can be a helpful story, but it’s vague and sometimes misleading. A more reliable AP-level idea is energy dispersal and the fact that systems naturally evolve toward macrostates with more accessible microstates.
• Forgetting kelvins: Any formula involving temperature in thermodynamics (especially $e_{Carnot}$ and $\Delta S = \frac{Q}{T}$) requires absolute temperature.

Exam Focus

• Typical question patterns:
  • Heat engine diagrams: given $Q_H$ and $Q_C$, find $W_{out}$ and $e$; or given $e$ and $Q_H$, find $Q_C$.
  • Carnot limit questions: compare a real engine’s efficiency to $e_{Carnot}$ for the same $T_H$ and $T_C$.
  • Entropy sign reasoning: determine whether $\Delta S_{univ}$ is positive, zero, or negative for a described process (often heat flow or a cycle).
• Common mistakes:
  • Using Celsius in $e_{Carnot} = 1 - \frac{T_C}{T_H}$ or in $\Delta S = \frac{Q}{T}$.
  • Assuming $\Delta S_{sys}$ must always increase; the correct statement is $\Delta S_{univ} \ge 0$ (a system’s entropy can decrease if surroundings increase more).
  • Treating efficiency as $\frac{Q_C}{Q_H}$ instead of $1 - \frac{Q_C}{Q_H}$, or mixing up $Q_H$ (in) and $Q_C$ (out).