1. Rate law for a single-step reaction

For the elementary gas-phase reaction $2A_{(g)} + B_{(g)} \rightarrow C_{(g)}$ the rate law is: $r = k P_A^2 P_B$ because for an elementary step, powers are equal to stoichiometric coefficients.

2. Initial rate

Initially, $P_A = 1.5\,\text{atm}, \qquad P_B = 0.7\,\text{atm}$ So, $r_1 = k(1.5)^2(0.7)$

3. Pressures after some time

When pressure of $C$ becomes $0.5\,\text{atm}$, use stoichiometry:

$2A + B \rightarrow C$

If $P_C$ increases by $0.5$ atm, then

• $A$ decreases by $2 \times 0.5 = 1.0$ atm
• $B$ decreases by $1 \times 0.5 = 0.5$ atm

Hence, $P_A = 1.5 - 1.0 = 0.5\,\text{atm}$ $P_B = 0.7 - 0.5 = 0.2\,\text{atm}$

Therefore, $r_2 = k(0.5)^2(0.2)$

4. Compute the ratio

$\frac{r_1}{r_2} = \frac{k(1.5)^2(0.7)}{k(0.5)^2(0.2)}$

Cancel $k$: $\frac{r_1}{r_2} = \frac{(1.5)^2 \cdot 0.7}{(0.5)^2 \cdot 0.2}$

Now, $(1.5)^2 = 2.25, \qquad (0.5)^2 = 0.25$ So, $\frac{r_1}{r_2} = \frac{2.25 \times 0.7}{0.25 \times 0.2} = \frac{1.575}{0.05} = 31.5$

5. Match with the required format

Given: $r_1 : r_2 = \underline{\hspace{1cm}} \times 10^{-1}$

Let the required integer be $n$. Then $n \times 10^{-1} = 31.5$ $n = 31.5 \times 10 = 315$

6. Final answer

The required nearest integer is: $\boxed{315}$