Animated Solution for Chemistry - Chemical Equilibrium: At $1990\text{ K}$ and $1\text{ atm}$ pressure, there are equal number of ${\text{Cl}}_2$ molecules and $\text{Cl}$ atoms in the reaction mixture. The value of $K_p$ for the reaction ${\text{Cl}}_2(g)\rightleftharpoons 2\text{Cl}(g)$ under the above conditions is $x\times 10^{-1}$. The value of $x$ is ......... . (Rounded off to the nearest integer)

Imagine you are standing inside a microscopic reaction vessel heated to a scorching $1990\text{ K}$. All around you, diatomic chlorine molecules (${\text{Cl}}_2$) are absorbing intense thermal energy and violently splitting apart into individual chlorine atoms ($\text{Cl}$). Simultaneously, these atoms are colliding and recombining back into molecules. This beautiful, dynamic dance is what we call chemical equilibrium.

The problem gives us a fascinating constraint: at equilibrium, the number of ${\text{Cl}}_2$ molecules exactly equals the number of $\text{Cl}$ atoms.

Why is this piece of information so powerful? Because in chemistry, the number of molecules is directly proportional to the number of moles (thanks to Avogadro's principle). If the molecules are equal in number, their moles must also be equal. Let's say we have $n$ moles of ${\text{Cl}}_2$ and $n$ moles of $\text{Cl}$.

Here is where the magic happens. The total number of moles in our container is simply $n+n=2n$.

The mole fraction ($\chi$) of any gas is its own moles divided by the total moles. For both our chlorine molecules and chlorine atoms, the math is identical:

They each make up exactly $50\%$ of the gas mixture!

Now that we have the mole fractions, we can find the partial pressures. Dalton's Law of Partial Pressures states that the partial pressure of a gas ($p_i$) is equal to its mole fraction multiplied by the total pressure ($P_{\text{total}}$).

We are given that the total pressure is $1\text{ atm}$. Let's calculate the partial pressures:

With our partial pressures in hand, we are ready to tackle the equilibrium constant, $K_p$. For the reaction ${\text{Cl}}_2(g)\rightleftharpoons 2\text{Cl}(g)$, the expression for $K_p$ is the partial pressure of the products raised to their stoichiometric coefficients, divided by the partial pressure of the reactants.

Notice the squared term in the numerator! That comes directly from the '2' in front of the $\text{Cl}$ atom in our balanced equation. This is a common place where students make silly mistakes, so always double-check your exponents.

Let's substitute our calculated partial pressures into the $K_p$ expression:

One of the $0.5$ terms in the numerator beautifully cancels out with the $0.5$ in the denominator, leaving us with:

The problem asks us to express this in the format $x\times 10^{-1}$. We can easily rewrite $0.5$ as $5\times 10^{-1}$.

Comparing this to the given expression, it is crystal clear that our unknown value $x$ is exactly $5$.