Relationship Kp = Kc(RT)^Δn

HOW to use it (units & value of R)

Three cases based on $\Delta n$ :

$\Delta n$	Relationship	Meaning
$\Delta n = 0$	$K_p = K_c$	equal moles of gas both sides
$\Delta n > 0$	$K_p > K_c$ (if $RT>1$ )	more gas moles produced
$\Delta n < 0$	$K_p < K_c$	gas moles consumed

Worked examples

Worked example Example 1 — Ammonia synthesis

$N_2(g) + 3H_2(g) \rightleftharpoons 2NH_3(g)$ at 400 K, $K_c = 0.5$ . Find $K_p$ (R = 0.0821).

Step 1: $\Delta n = 2 - (1+3) = -2$ . Why? Count gas moles: 2 product, 4 reactant. Step 2: $K_p = K_c(RT)^{\Delta n} = 0.5\times(0.0821\times400)^{-2}$ Why? Substitute directly; note negative exponent because gas moles decrease. Step 3: $RT = 32.84$ , so $K_p = 0.5 / (32.84)^2 = 0.5/1078 = 4.6\times10^{-4}$ . Why smaller? $\Delta n<0$ and $RT>1$ , so $(RT)^{\Delta n}<1$ ⇒ $K_p<K_c$ . ✔

Common mistakes

Flashcards

What is the relationship between Kp and Kc?

K_p = K_c(RT)^{\Delta n}

Define

\Delta n

K_p=K_c(RT)^{\Delta n}

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

Which gas law is used to derive

K_p=K_c(RT)^{\Delta n}

Ideal gas law, via

p_X = [X]RT

When does

K_p = K_c

When

\Delta n = 0

(equal gas moles on both sides), since

(RT)^0=1

For

N_2+3H_2\rightleftharpoons 2NH_3

, what is

\Delta n

2-4 = -2

Which species are counted in

\Delta n

Only gaseous species; solids and liquids are excluded.

What must the temperature unit be?

Kelvin.

Value of R for atm and mol/L?

0.0821\ \text{L atm K}^{-1}\text{mol}^{-1}

\Delta n>0

and

RT>1

, is

K_p

bigger or smaller than

K_c

Bigger (

K_p>K_c

Derive

p_X

in terms of concentration.

p_X V=n_XRT \Rightarrow p_X=(n_X/V)RT=[X]RT

Recall Feynman: explain to a 12-year-old

Imagine a reaction happening inside a box full of gases. You can describe "how crowded" each gas is in two ways: by how many molecules sit in each litre (concentration) or by how hard they push on the walls (pressure). The ideal gas law says these two are the same thing scaled by $RT$ : push = crowd × $RT$ . So the two ways of writing the equilibrium constant differ only by some $RT$ 's. How many $RT$ 's? Exactly the difference in the number of gas molecules between the right and left side of the arrow. If both sides have the same number of gas molecules, the $RT$ 's cancel and the two constants are equal!

Connections

Law of Chemical Equilibrium — where $K_c$ is defined.
Equilibrium Constant Kp — partial-pressure form.
Ideal Gas Law PV=nRT — the engine of this derivation.
Partial Pressure and Dalton's Law — meaning of $p_X$ .
Le Chatelier's Principle — effect of pressure changes on $\Delta n\neq0$ reactions.
Gibbs Free Energy and K — $\Delta G^\circ = -RT\ln K$ .

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, gas-phase equilibrium ko hum do tareeke se likh sakte hain: concentration ke through ( $K_c$ ) ya partial pressure ke through ( $K_p$ ). Dono same equilibrium ko describe karte hain, toh inke beech ek relation hona chahiye. Ye relation aata hai ideal gas law se — kyunki $pV = nRT$ ko rearrange karein toh $p_X = (n_X/V)RT = [X]RT$ ban jaata hai. Matlab har gas ka pressure uski concentration ka $RT$ guna hai.

Ab jab is $p_X = [X]RT$ ko $K_p$ ke expression mein daal do, toh saare concentration terms milkar $K_c$ ban jaate hain, aur $RT$ ke powers alag nikal aate hain. $RT$ ka power exactly $\Delta n$ hota hai — yaani gaseous products ke moles minus gaseous reactants ke moles. Final formula: $K_p = K_c(RT)^{\Delta n}$ .

Do cheezein hamesha yaad rakho: (1) $T$ hamesha Kelvin mein, Celsius mat daalna. (2) $\Delta n$ mein sirf gases count hoti hain — solid aur liquid ko chhod do. Jaise $CaCO_3(s) \to CaO(s) + CO_2(g)$ mein $\Delta n = 1$ hai, sirf $CO_2$ gas hai. Agar $\Delta n = 0$ ho (dono taraf equal gas moles), toh $K_p = K_c$ ho jaata hai kyunki $(RT)^0 = 1$ .

Ye topic important hai kyunki numericals mein aksar $K_c$ diya hoga aur $K_p$ poocha jaayega (ya ulta), aur thermodynamics mein $\Delta G^\circ = -RT\ln K$ ke saath link hota hai. Bas formula samajh ke derive karna aata ho, ratta maarne ki zaroorat nahi.

WHAT are we relating?

HOW to use it (units & value of R)

Worked examples

Common mistakes

Flashcards

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Equilibrium

Connections