Given,
\(A_{2}(g)+B_{2}(g) \rightleftharpoons 2 A B(g)\)
1 mole of \(A_{2}\) reacts with 1 mole of \(B_{2}\) will produce 2 moles of \(A B\). Given, the value of equilibrium constant is \(K_{c}=4\). The \(K_{c}\) means the value of the compounds are taken in concentrations.
In the reaction given above, there are 2 moles of \(A_{2}\) and 4 moles of \(B_{2}\). Initially, the moles of \(A\) are 2 , moles of \(B\) are 4 , and mole of \(A B\) is 0 . After time \(t\), the moles of \(A\) will be \(2-x\), the moles of \(B\) will be \(4-x\), and the moles of \(A B\) will be \(2 x\). Given the volume is 1 liter of \(A_{2}\) and 3 liters of \(B_{2}\), so the total volume of the system will be \(3+1=4\) liters.
Total volume \(=1+3=4\) litre
\({[\mathrm{AB}]=\frac{2 x}{4}=\frac{x}{2} }\)
\({\left[\mathrm{~B}_{2}\right]=\frac{4-x}{4} }\)
\({\left[\mathrm{~A}_{2}\right]=\frac{2-x}{4} }\)
At equilibrium,
\(K_{c}=\frac{[A B]^{2}}{\left[A_{2}\right]\left[B_{2}\right]}\)
\(K_{\mathbf{c}}=\frac{\left(\frac{x}{2}\right)^{2}}{\left(\frac{4-x}{4}\right)\left(\frac{2-x}{4}\right)}\)
\(\Rightarrow 4=\frac{4 x^{2}}{(2-x)(4-x)}\)
\(\Rightarrow x^{2}=(4-x)(2-x)\)
\(\Rightarrow x^{2}=x^{2}-6 x+8\)
\(\Rightarrow 6 x=8\)
\(\Rightarrow x=\frac{4}{3}\)
\(\Rightarrow x=\frac{32}{24}=1.33\) mole
The concentration of \(A B\) is \(\frac{x}{2}\), putting the value of \(x\)
\([A B]=\frac{2}{3}=0.66 M\)