Reaction $$A(g) \rightarrow 2B(g) + C(g)$$ is a first order reaction. It was started with pure A.

Which of the following option is incorrect?

The stoichiometry is $$A(g) \rightarrow 2B(g)+C(g)$$ and the reaction is first-order in $$A$$.

Let the initial partial pressure of $$A$$ be $$P_0$$ mm Hg.
At any time $$t$$, let $$x$$ mm Hg of $$A$$ have decomposed.

Pressures at time $$t$$:
  • Unreacted $$A$$ : $$P_A = P_0 - x$$
  • Product $$B$$ : $$2x$$ (because 2 mol $$B$$ per mol $$A$$)
  • Product $$C$$ : $$x$$
Hence total pressure at time $$t$$ is

$$P_t = (P_0 - x) + 2x + x = P_0 + 2x$$ $$-(1)$$

Case 1: Pressure at $$t \rightarrow \infty$$

All of $$A$$ is consumed, so $$x = P_0$$.
Using $$(1)$$ with $$P_\infty = 240$$ mm Hg:

$$240 = P_0 + 2P_0 = 3P_0 \; \Rightarrow \; P_0 = 80 \text{ mm Hg}$$

Thus the initial pressure is 80 mm Hg  →  Option A is correct.

Case 2: Pressure after 10 min

Given $$P_{10} = 160$$ mm Hg.
Insert $$P_0 = 80$$ mm Hg into $$(1)$$:

$$160 = 80 + 2x \; \Rightarrow \; x = 40 \text{ mm Hg}$$

Partial pressure of $$A$$ after 10 min:

$$P_A = P_0 - x = 80 - 40 = 40 \text{ mm Hg}$$

Hence Option D is correct.

Case 3: Rate constant

For a first-order reaction, $$k = \dfrac{1}{t}\,\ln\!\left(\dfrac{P_0}{P_A}\right)$$.

Substituting $$P_0 = 80$$ mm Hg, $$P_A = 40$$ mm Hg and $$t = 10$$ min:

$$k = \dfrac{1}{10}\,\ln\!\left(\dfrac{80}{40}\right) = \dfrac{1}{10}\,\ln 2$$

$$\ln 2 \approx 0.693$$, so

$$k = \dfrac{0.693}{10} = 0.0693 \text{ min}^{-1}$$

The value quoted in Option C (1.693 min$$^{-1}$$) is therefore wrong.

Case 4: Does the reaction go to completion?

For a first-order kinetics equation, $$[A] = [A]_0 e^{-kt}$$ never becomes exactly zero in a finite time; it only approaches zero as $$t \rightarrow \infty$$. Hence one may say the reaction “never actually reaches completion.” This matches Option B.

Summary:
Option A - correct (initial pressure 80 mm Hg)
Option B - correct (strictly never reaches completion in finite time)
Option C - incorrect (true value 0.0693 min$$^{-1}$$)
Option D - correct (40 mm Hg of $$A$$ after 10 min)

The incorrect statement is Option C.