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JEE Advanced levelOne or more correct optionsDifficulty 3 of 3ck152

The first-order reaction $A (g) 3 B (g)$ starts with pure A at pressure $P_{0}$ in a rigid vessel at constant temperature. $P_{t}$ is the total pressure. Which statements are correct?

A $P_{t} = P_{0} + 2 x$ , where $x$ is the pressure of A that has reacted
B $k = \frac{1}{t} ln \frac{2 P _{0}}{3 P _{0} - P _{t}}$
CAfter two half-lives $P_{t} = 2.25 P_{0}$
DAfter one half-life $P_{t} = 2 P_{0}$

Answer: A, B, D

Each A gives 3 B, so

P_{t} = P_{0} - x + 3 x = P_{0} + 2 x

and

P_{\infty} = 3 P_{0}

. After one half-life

x = P_{0} /2

giving

2 P_{0}

. After two half-lives

x = 3 P_{0} /4

giving

2.5 P_{0}

, not

2.25 P_{0}

. Also

p_{A} = P_{0} - (P_{t} - P_{0}) /2 = (3 P_{0} - P_{t}) /2

, so

k = \frac{1}{t} ln \frac{2 P _{0}}{3 P _{0} - P _{t}}

.

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