Question

a tiny sample of an aqueous solution of two substances r and p is sketched below, as if it was under an imaginary microscope so powerful that individual molecules could be seen. (the water molecules are not shown.)

r and p can interconvert. that is, r can turn into p, and p can turn back into r:

$$ r ( a q ) ightleftharpoons p ( a q ) $$

$$ k = \frac { 3 } { 2 } $$

the equilibrium constant k for this equilibrium is $$ \frac { 3 } { 2 } $$. predict the number of r and p molecules in this sample when the interconversion reaches equilibrium.

number of r molecules
number of p molecules:

Explanation:

Step1: Count the initial number of molecules

There are \(6\) \(R\) molecules and \(6\) \(P\) molecules initially. The total number of molecules \(n = 6 + 6=12\).

Step2: Set up the equation for equilibrium

Let \(x\) be the number of \(R\) molecules that convert to \(P\). At equilibrium, the number of \(R\) molecules is \(6 - x\) and the number of \(P\) molecules is \(6 + x\). The equilibrium constant \(K=\frac{[P]}{[R]}\) (since concentration is proportional to number of molecules in the same - volume sample). Given \(K = \frac{3}{2}\), we have \(\frac{6 + x}{6 - x}=\frac{3}{2}\).

Step3: Solve the equation

Cross - multiply: \(2(6 + x)=3(6 - x)\).
Expand: \(12+2x = 18-3x\).
Add \(3x\) to both sides: \(12 + 2x+3x=18-3x + 3x\), so \(12 + 5x=18\).
Subtract \(12\) from both sides: \(5x=18 - 12\), \(5x = 6\), \(x=\frac{6}{5}=1.2\) (This is wrong, we should use the ratio of whole - number of molecules).

Another way: Since \(K=\frac{[P]}{[R]}=\frac{3}{2}\), let the number of \(R\) molecules be \(2y\) and the number of \(P\) molecules be \(3y\). And \(2y + 3y=12\) (total number of molecules).

Step4: Solve for \(y\)

\(5y = 12\) (wrong, because we should consider whole - number ratios). The ratio \(P:R = 3:2\). The total number of parts in the ratio is \(3 + 2=5\). But \(12\) is not divisible by \(5\). Let's consider the fact that the ratio of the number of \(P\) and \(R\) molecules at equilibrium is \(3:2\). The closest whole - number combination for a total of \(12\) molecules:
If \(R = 4\) and \(P = 6\), \(K=\frac{6}{4}=\frac{3}{2}\)

Answer:

number of \(R\) molecules: \(4\)
number of \(P\) molecules: \(6\)