What are the values of $$\frac{K_p}{K_c}$$ for the following reactions at 300 K respectively?
(At 300 K, RT = 24.62 dm$$^2$$ atm mol$$^{-1}$$)
$$N_2(g) + O_2(g) \rightleftharpoons 2NO(g)$$
$$N_2O_4(g) \rightleftharpoons 2NO(g)$$
$$N_2(g) + 3H_2(g) \rightleftharpoons 2NH_3(g)$$

For gaseous reactions we have the relation between the two equilibrium constants

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

where $$\Delta n =(\text{total gaseous moles of products})-(\text{total gaseous moles of reactants})$$ and $$RT$$ has the value given in the question, $$RT = 24.62\ \text{dm}^3\ \text{atm mol}^{-1}$$ at 300 K.

Re-writing the relation to isolate the required ratio,

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

Now we evaluate this ratio for each reaction one by one, carefully calculating $$\Delta n$$ each time.

First reaction : $$N_2(g)+O_2(g)\rightleftharpoons 2NO(g)$$

The number of gaseous moles on the left is $$1+1=2$$ while on the right it is $$2$$. Hence $$\Delta n = 2-2 = 0$$.

Substituting $$\Delta n=0$$ gives

$$\frac{K_p}{K_c}=(RT)^0 = 1$$

Second reaction : $$N_2O_4(g)\rightleftharpoons 2NO(g)$$

Moles on the left = $$1$$; moles on the right = $$2$$. Therefore $$\Delta n = 2-1 = 1$$.

So

$$\frac{K_p}{K_c}=(RT)^1 = RT = 24.62\ \text{dm}^3\ \text{atm mol}^{-1}$$

Third reaction : $$N_2(g)+3H_2(g)\rightleftharpoons 2NH_3(g)$$

Moles on the left = $$1+3 = 4$$; moles on the right = $$2$$, giving $$\Delta n = 2-4 = -2$$.

Hence

$$\frac{K_p}{K_c}=(RT)^{-2}=\frac{1}{(RT)^2}$$

We now square the given numerical value of $$RT$$:

$$RT = 24.62$$ gives $$ (RT)^2 = (24.62)^2 = 606.1\ \text{dm}^6\ \text{atm}^2\ \text{mol}^{-2}$$ (multiplying 24.62 by 24.62 step-by-step yields 606.1444, which we may round to 606.1).

Therefore

$$\frac{K_p}{K_c}= \frac{1}{606.1}=1.65\times10^{-3}\ \text{dm}^{-6}\ \text{atm}^{-2}\ \text{mol}^2$$

Collecting the three results in the order asked, we have

$$1,\; 24.62\ \text{dm}^3\ \text{atm mol}^{-1},\; 1.65\times10^{-3}\ \text{dm}^{-6}\ \text{atm}^{-2}\ \text{mol}^2$$

These values exactly match the figures presented in Option C.

Hence, the correct answer is Option C.