1. Given reaction

$2\,\mathrm{N_2O_5(g)} \rightarrow 2\,\mathrm{N_2O_4(g)} + \mathrm{O_2(g)}$

Initial total pressure at $t=0$ is only due to $\mathrm{N_2O_5}$:

$P_0 = 0.6\ \text{atm}$

At constant volume and temperature, pressure is proportional to moles.

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2. Kinetics of decomposition

Thermal decomposition of $\mathrm{N_2O_5}$ is first order, so

$k = \frac{1}{t}\ln\frac{P_0(\mathrm{N_2O_5})}{P_t(\mathrm{N_2O_5})}$

Given:

$k = 4.606\times 10^{-3}\ \text{s}^{-1}, \qquad t=100\ \text{s}$

So,

$kt = 4.606\times 10^{-3}\times 100 = 0.4606$

Hence,

$\frac{P_t(\mathrm{N_2O_5})}{P_0} = e^{-kt} = e^{-0.4606}$

Using $0.4606 \approx 2.303\times 0.2$,

$e^{-0.4606} = 10^{-0.2} \approx 0.631$

Therefore,

$P_t(\mathrm{N_2O_5}) = 0.6\times 0.631 = 0.3786\ \text{atm}$

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3. Relating decomposition to total pressure

Let initial moles of $\mathrm{N_2O_5}$ be $a$.

If $y$ moles decompose, then from

$2\mathrm{N_2O_5} \rightarrow 2\mathrm{N_2O_4} + \mathrm{O_2}$

the total moles become

$n_t = (a-y) + y + \frac{y}{2} = a + \frac{y}{2}$

Thus total pressure at time $t$ is

$P_t^{\text{total}} = P_0\left(1+\frac{y}{2a}\right)$

Now fraction undecomposed is

$\frac{a-y}{a} = 0.631$

So,

$\frac{y}{a} = 1-0.631 = 0.369$

Hence,

$P_t^{\text{total}} = 0.6\left(1+\frac{0.369}{2}\right)$

$=0.6(1+0.1845)=0.6\times 1.1845 = 0.7107\ \text{atm}$

So,

$x = 0.7107\ \text{atm} = 710.7\times 10^{-3}\ \text{atm}$

Nearest integer:

$x = 711$

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4. Comparison with stored answer

Stored correct answer = $897$

Derived answer = $711$

These do not match.

The stored answer would be inconsistent with the standard first-order decomposition relation and stoichiometric pressure increase for

$2\mathrm{N_2O_5} \rightarrow 2\mathrm{N_2O_4} + \mathrm{O_2}$

If instead one incorrectly used $k=4.606\times 10^{-2}\,\mathrm{s^{-1}}$, then after $100\,$s almost complete decomposition would occur and pressure would approach $0.9\,$atm, i.e. $900\times 10^{-3}$ atm, close to $897$. So the stored answer likely comes from a typo in the rate constant exponent. With the given answer format and stored key, the intended value appears to be based on $k=4.606\times 10^{-2}\,\mathrm{s^{-1}}$, but with the physically correct reading for this problem setup, the answer is $711$.