Initially a gas of diatomic molecules is contained in a cylinder of volume $$V_1$$ at a pressure $$P_1$$ and temperature $$250\,\text{K}$$. Assuming that $$25\%$$ of the molecules get dissociated causing a change in number of moles. The pressure of the resulting gas at temperature $$2000\,\text{K}$$, when contained in a volume $$2V_1$$ is given by $$P_2$$. The ratio $$P_2/P_1$$ is ___

We begin with the ideal-gas equation, which is written as $$PV = nRT,$$ where $$P$$ is the pressure, $$V$$ is the volume, $$n$$ is the number of moles, $$R$$ is the universal gas constant and $$T$$ is the absolute temperature.

Initially the gas is diatomic, so let the initial number of moles be $$n$$. The initial state is described by

$$P_1V_1 = nR(250\,\text{K}).$$

Now 25 % of the diatomic molecules dissociate according to the reaction

$$\text{A}_2 \;\longrightarrow\; 2\text{A}.$$

Out of the original $$n$$ moles, the fraction that dissociates is $$0.25n$$. Each mole of dissociating diatomic molecules produces two moles of monatomic gas, so the number of moles of atoms produced is

$$2 \times 0.25n \;=\; 0.50n.$$

The number of moles of diatomic molecules that remain undissociated is

$$n - 0.25n \;=\; 0.75n.$$

Thus the total number of moles after dissociation becomes

$$n_2 = 0.75n + 0.50n = 1.25n.$$

This dissociated mixture is now kept in a volume that is twice the original, namely $$2V_1,$$ and its temperature is raised to $$2000\,\text{K}$$. Applying the same ideal-gas equation to the final state gives

$$P_2(2V_1) = n_2R(2000\,\text{K}).$$

Substituting $$n_2 = 1.25n$$ we write

$$P_2(2V_1) = 1.25nR \times 2000.$$

Now we isolate $$P_2$$:

$$P_2 = \frac{1.25nR \times 2000}{2V_1}.$$

For the ratio $$\dfrac{P_2}{P_1}$$ we divide this expression by the earlier expression for $$P_1$$. From the initial state we have

$$P_1 = \frac{nR \times 250}{V_1}.$$

Taking the ratio,

$$\frac{P_2}{P_1} = \frac{\dfrac{1.25nR \times 2000}{2V_1}}{\dfrac{nR \times 250}{V_1}}.$$

The factors $$n$$, $$R$$ and $$V_1$$ cancel out completely, leaving

$$\frac{P_2}{P_1} = \frac{1.25 \times 2000}{2 \times 250}.$$

Calculating the numerator first, $$1.25 \times 2000 = 2500$$. Dividing by 2 gives $$1250$$. Finally,

$$\frac{P_2}{P_1} = \frac{1250}{250} = 5.$$

Hence, the correct answer is Option 5.