Enthalpy H = U + PV; ΔH for reactions at constant P: Foundations

Before you can trust the formula $H = U + P V$ , you must be able to read every mark in it. We will meet each symbol one at a time, give it plain words, draw the picture it stands for, and say why the topic cannot live without it. No symbol is used until it has been defined.

The stage: system, surroundings, boundary

Everything in thermodynamics starts by drawing a line around the piece of the universe we care about.

Figure — Enthalpy H = U + PV; ΔH for reactions at constant P

The picture also shows the boundary is open at the top: air can push down but the chemicals are free to swell upward. Hold that image — it is the whole reason enthalpy exists.

The symbol $U$ — internal energy

We link the deeper story of this quantity to Internal Energy U and First Law.

The symbols $q$ and $w$ — heat and work crossing the boundary

These are the two ways energy sneaks across the boundary.

Why the topic needs $q$ and $w$ : enthalpy is invented precisely to make one of these two — the heat $q$ — easy to read off. Everything hinges on splitting $Δ U$ into a heat part and a work part.

The symbols $P$ and $V$ — pressure and volume

Now the two symbols that turn "work" into something we can compute.

Building $P V$ into a new bookkeeping quantity: $H$

Here is the trick the whole topic rests on. We now add one crucial assumption: the process is slow and open, so at every instant the inside pressure balances the outside pressure, $P_{ext} = P_{system} = P (constant) .$ Only because these two are equal can we drop the "ext" label and write a single $P$ . Substituting $w = - P Δ V$ into the First Law:

$Δ U = q_{P} - P Δ V ⟹ q_{P} = Δ U + P Δ V$

The subscript $P$ on $q_{P}$ is a reminder: "this heat was measured while pressure stayed constant." Now group the terms so each side is (energy) + (pressure×volume):

$q_{P} = (U + P V)_{final} - (U + P V)_{initial}$

Why bundle them? Because now a thermometer in an open cup measures $q_{P}$ directly, and that number is $Δ H$ with no separate work correction to worry about.

The sign of $Δ H$ — the picture

$n$ , $R$ , $T$ and the $Δ n_{gas} R T$ bridge

To connect $Δ H$ back to the honest $Δ U$ for gas reactions we need three more symbols.

Why the topic needs this: a bomb calorimeter holds volume fixed and measures $Δ U$ ; an open cup holds pressure fixed and measures $Δ H$ . This one line converts between the two. See Calorimetry.

How the pieces feed the topic

Read the map below bottom-up as a build order: start at the top boxes (things you already have — a system, its internal energy, heat and work), follow the arrows down, and each arrow means "this is needed before the next box makes sense." The two ideas First Law and PV work flow together into the definition of $H$ , which then gives $Δ H = q_{P}$ and finally $Δ H$ for reactions. If any single box feels shaky, jump back to that section above before reading the parent topic.

Equipment checklist

Cover the right side and check you can say each one.

What does a boundary do in thermodynamics?

It separates the system from the surroundings so we can say what "crosses" it.

What is

U

in plain words?

The total energy hidden inside molecules — their jiggling plus bond and attraction energy.

What is the First Law?

Δ U = q + w

: change in internal energy equals heat in plus work in.

What does

Δ

mean?

Final value minus initial value — the change in a quantity.

What sign convention do

q

and

w

use?

Energy into the system is positive; energy out is negative.

Why is

P V

work written

w = - P Δ V

Expanding gas spends energy pushing the surroundings, so the system loses energy (minus sign).

Why may we replace

P_{ext}

with a single

P

Because at constant pressure the outside and inside pressures balance,

P_{ext} = P_{system} = P

What does

\equiv

mean in

H \equiv U + P V

"Is defined as" — a chosen name, not a derived result.

At constant pressure,

Δ H

equals what?

The heat exchanged,

q_{P}

Exothermic means what sign of

Δ H

Negative — heat exits, surroundings warm.

What is

R

and its value?

The gas constant,

8.314 J mol^{- 1} K^{- 1}

How do you convert

Δ H

Δ U

for gases?

Δ H = Δ U + Δ n_{gas} R T

Ready? Then continue to the parent: the full topic, and the prerequisite depth in Internal Energy U and First Law and PV Work and Expansion.