Question

At $373K$ , a gaseous reaction $A \to 2B + C$ is found to be first-order. Starting with pure $A$ , the total pressure at the end of $10\min$ was $176mmHg$ and after a long time when $A$ was completely dissociated, it was $270mmHg$ . The pressure of $A$ at the end of $10\min$ was
A.$94mmHg$
B.$47mmHg$
C.$43mmHg$
D.$90mmHg$

Answer 1

Differential rate laws are used to describe the events occurring on a molecular level during a reaction whereas integrated rate laws are used for determining order of reaction and the value of rate constant during experimental conditions.

Complete step by step answer:
The given reaction is a first-order reaction. A reaction whose rate is dependent on the concentration of only one reactant is called a first-order reaction. Now in the given reaction,
$A \to 2B + C$

At time $= 0$	$x$	$0$	$0$
At t $= 10\min$	$x - y$	$2y$	$y$

Let $x$ be the pressure of $A$ . Then the pressure of $B$ and $C$will be zero since at zero time there will be no product formation.
Then after $10\min$ , if pressure of product $C$ is considered as $y$ , then pressure of $B$ wil be $2y$ and that of $A$ will be $x - y$ . It is given that the pressure after $10\min$ is $176mmHg$ .
So, total pressure $= (x - y) + 2y + y = 176mmHg$
$x + 2y = 176mmHg$ ……(i)
Now it is given that after $100\min$ the pressure when$A$ is completely dissociated is $270mmHg$ .
So, $A \to 2B + C$

At t $= 100\min$	$0$	$2x$	$x$

Since $A$ is completely dissociated the pressure of $C$ will be equal to the initial pressure of $A$ which will be $x$ . and that of $B$ will be $2x$ .
So, $0 + 2x + x = 270mmHg$
$3x = 270mmHg$
$x = 90mmHg$
Substituting this value in equation (i),
$x + 2y = 176mmHg$
$90 + 2y = 176$
$2y = 176 - 90$
$2y = 86$
$y = 43mmHg$
Now we know that at the end of $10\min$ the pressure of $A$ is $x - y$ .
Substituting values of $x$ and $y$ we get,
Pressure of $A$ $= x - y = 90 - 43 = 47mmHg$
So the correct option is B.

Note: The rate law equation for first order reaction is $k = \dfrac{{2.303}}{t}\log \dfrac{a}{{a - x}}$ Where $k$ is reaction rate coefficient, $t$ is the time, $a$ is the initial concentration of reactant and $a - x$ is concentration of reactant left behind after time $t$ .
-The unit of $k$ is $tim{e^{ - 1}}$ .
-The unit of $k$ varies for every order of reaction.