Relationship Kp = Kc(RT)^Δn: Foundations — Chemistry deep dive

Before we can trust the formula $K_{p} = K_{c} (R T)^{Δ n}$ , we must earn every single symbol in it. This page is the toolbox. We open every drawer, name every tool, draw the picture it stands for, and say why the topic can't work without it. Nothing here assumes you've seen chemistry notation before.

1. The reaction arrow $⇌$ and the letters $a, b, c, d, A, B, C, D$

Figure 1 — the two teams. The left lavender box holds the reactants ( $a A + b B$ ): four big lavender marbles (species $A$ ) and three small coral marbles (species $B$ ). The right mint box holds the products ( $c C + d D$ ). The two arrows between them — coral pointing right ("forward"), lavender pointing left ("backward") — are the double arrow $⇌$ drawn as motion: both changes happen at once. Every symbol later in this page is really a question about these two teams.

2. Concentration $[X]$ — how crowded a gas is

Here $n_{X}$ = number of moles of $X$ (a mole is just "a fixed huge count of molecules" — like a "dozen" but enormous). $V$ = volume of the container in litres (L).

3. Partial pressure $p_{X}$ — how hard one gas pushes

Figure 2 — same box, two descriptions. The left (butter) panel describes the gas by crowding: count the coral marbles and divide by the litres of box → concentration $[X]$ , which feeds $K_{c}$ . The right (lavender) panel describes the same gas by pushing: the little grey arrows are molecules hammering the walls → partial pressure $p_{X}$ , which feeds $K_{p}$ . The coral caption between them, $p_{X} = [X] \times R T$ , is the bridge that turns one description into the other — the single equation the whole topic rests on.

4. The ideal gas law $p V = n R T$ — the bridge

5. The gas constant $R$ and temperature $T$

Definition

$R$ — the unit-matching number $R$ is a fixed number that makes $p V = n R T$ balance. Its numerical value depends on which units you use — and crucially, the volume $V$ inside $p V = n R T$ must use the same volume unit as your concentration (litres with mol L⁻¹, cubic metres with mol m⁻³). Never mix L and m³.

Pressure unit	Concentration / volume unit	$R$
atm	mol L⁻¹ (so $V$ in L)	$0.0821 L atm K^{- 1} mol^{- 1}$
bar	mol L⁻¹ (so $V$ in L)	$0.08314 L bar K^{- 1} mol^{- 1}$ ( $\approx 0.083$ )
Pa (SI)	mol m⁻³ (so $V$ in m³)	$8.314 J K^{- 1} mol^{- 1}$

6. The equilibrium constants $K_{c}$ and $K_{p}$

7. The exponent $\sum$ and $Δ n$

Figure 3 — counting $Δ n$ . Using $N_{2} (g) + 3 H_{2} (g) ⇌ 2 N H_{3} (g)$ : the left lavender box counts the reactant gas moles ( $1 + 3 = 4$ marbles), the right mint box counts the product gas moles ( $2$ marbles). Subtract product minus reactant: $Δ n = 2 - 4 = - 2$ . The coral arrow reminds you which side is subtracted from which — always products minus reactants, never the reverse.

8. The full substitution, every $R T$ tracked

Words like "each species drags one factor of $R T$ " are only convincing if we watch it happen. Here is the complete algebra, starting from $K_{p}$ and using the bridge $p_{X} = [X] R T$ from Section 4.

9. Putting the toolbox together

Every tool now has a name, a picture, and a reason:

Symbol	Plain meaning	Why the topic needs it
$⇌$	reaction runs both ways (Fig 1)	there's a balance to have a constant for
$[X]$	concentration / crowding (Fig 2, left)	builds $K_{c}$
$p_{X}$	partial pressure / pushing (Fig 2, right)	builds $K_{p}$
$p V = n R T$	push = crowd × $R T$ (ideal gas)	the bridge $p_{X} = [X] R T$
$R$	unit-matching number	scales push↔crowd
$T$	temperature (kelvin)	appears in $R T$
$K_{c}, K_{p}$	equilibrium constants	the two things we relate
$Δ n$	change in gas moles (Fig 3)	the exponent of $R T$

Once these are in hand, the parent formula $K_{p} = K_{c} (R T)^{Δ n}$ is nothing more than "swap push for crowd, collect the leftover $R T$ 's" — exactly the five steps of Section 8.

Prerequisite map

Equipment checklist

What does the double arrow

⇌

mean?

The reaction runs forward and backward at the same time, reaching a balance (equilibrium).

What is

[X]

and its unit?

Molar concentration

n_{X} / V

, moles per litre (mol L⁻¹) — how crowded gas

X

is.

What is a partial pressure

p_{X}

The pressure gas

X

would exert alone, ignoring the other gases in the mixture.

State the ideal gas law and the bridge it gives (and its assumption).

p V = n R T

for an ideal gas, giving

p_{X} = (n_{X} / V) R T = [X] R T

Why does

R

have different numeric values, and what must

V

match?

It matches the chosen pressure/volume units (atm+L, bar+L, Pa+m³);

V

p V = n R T

must use the same volume unit as the concentration.

Convert 27 °C to the temperature used in the gas law.

27 + 273.15 = 300.15 K

(often rounded to

300 K

Write the general form of

K_{c}

, and what is left out.

Products over reactants each to its coefficient,

\frac{[ C ] ^{c} [ D ] ^{d}}{[ A ] ^{a} [ B ] ^{b}}

; pure solids and liquids are omitted (activity = 1).

Define

Δ n

Moles of gaseous products minus moles of gaseous reactants (products − reactants).

Which species are excluded from

Δ n

Solids and liquids — only gases are counted.

Why does

Δ n

become the exponent of

R T

Each gas swaps

p_{X}

for

[X] R T

; the

R T

's collect as (product powers − reactant powers) =

Δ n

Connections

Law of Chemical Equilibrium — where the products-over-reactants form of $K$ comes from.
Equilibrium Constant Kp — the partial-pressure constant defined in Section 6.
Ideal Gas Law PV=nRT — the bridge $p_{X} = [X] R T$ .
Partial Pressure and Dalton's Law — meaning of $p_{X}$ in a mixture.
Le Chatelier's Principle — why $Δ n$ controls response to pressure.
Gibbs Free Energy and K — deeper meaning of $K$ via $Δ G^{\circ} = - R T ln K$ .