For a first order reaction A(g) → 2B(g) + C(g) at constant volume and 300 K, the total pressure at the beginning (t = 0) and at time t are P0 and Pt, respectively. Initially, only A is present with concentration [A]0, and t1/3 is the time required for the partial pressure of A to reach 1/3rd of its initial value. The correct option(s) is (are) (Assume that all these gases behave as ideal gases)

Engineering

Chemistry

Calculation of First Order Rate Constant using Different Parameters

Question

For a first order reaction A(g) → 2B(g) + C(g) at constant volume and 300 K, the total pressure at the beginning (t = 0) and at time t are P₀ and P_t, respectively. Initially, only A is present with concentration [A]₀, and t_1/3 is the time required for the partial pressure of A to reach 1/3^rd of its initial value. The correct option(s) is (are) (Assume that all these gases behave as ideal gases)

A(g) → 2B(g) + C(g)

t = 0 P₀ 0 0

t = t P₀ – P 2P P P_t = P₀ + 2P

$Kt = ln \frac{P_{0}}{P_{0} - (\frac{P_{t} - P_{0}}{2})} = = \ln \frac{2 P_{0}}{3 P_{0} - P_{t}}$ = ln2P₀ – ln(3P₀–P_t)

ln(3P₀– P_t) = – Kt + ln2P₀

${Kt}_{1 / 3} = ln \frac{1}{1 / 3} = ln3$

$t_{1 / 3} = \frac{\ln 3}{K} \Rightarrow t_{13}$ is independent of initial concentration

Understanding the First Order Reaction and Pressure Changes

We are given a first order reaction: $A (g) \to 2 B (g) + C (g)$ . The reaction occurs at constant volume and 300 K. The total pressure at time t=0 is $P_{0}$ and at time t is $P_{t}$ . Initially, only A is present. $t_{1 / 3}$ is defined as the time required for the partial pressure of A to drop to one-third of its initial value, i.e., $P_{A} = P_{0} / 3$ .

Since the reaction is first order, the rate constant k is given by: $k = \frac{1}{t} \ln \frac{[A]_{0}}{[A]}$ . For concentrations, we can replace them with partial pressures for gaseous reactions at constant volume because pressure is proportional to concentration (from ideal gas law, P = (n/V)RT, so P ∝ concentration).

Step 1: Relate Concentration to Pressure

Let the initial pressure of A be $P_{0}$ . Therefore, initial concentration ${[A]}_{0} \propto P_{0}$ .

At any time t, let the decrease in partial pressure of A be x. So, partial pressure of A at time t, $P_{A} = P_{0} - x$ .

From the stoichiometry $A \to 2 B + C$ :

Pressure due to B = 2x
Pressure due to C = x

Therefore, the total pressure at time t is: $P_{t} = P_{A} + P_{B} + P_{C} = (P_{0} - x) + 2 x + x = P_{0} + 2 x$

From this, we can solve for x: $2 x = P_{t} - P_{0}$ so $x = \frac{P_{t} - P_{0}}{2}$ .

Therefore, the partial pressure of A at time t is: $P_{A} = P_{0} - x = P_{0} - \frac{P_{t} - P_{0}}{2} = \frac{3 P_{0} - P_{t}}{2}$

Step 2: Apply the First Order Kinetics Equation

The first order rate law in terms of partial pressure is: $k = \frac{1}{t} \ln \frac{P_{0}}{P_{A}}$

Substituting the expression for $P_{A}$ : $k = \frac{1}{t} \ln (\frac{P_{0}}{\frac{3 P_{0} - P_{t}}{2}}) = \frac{1}{t} \ln (\frac{2 P_{0}}{3 P_{0} - P_{t}})$

This is a general expression relating $P_{t}$ and $P_{0}$ for this reaction.

Step 3: Find the Expression for the Rate Constant k

The rate constant for a first order reaction is a fixed value. From the standard formula, we know $k = \frac{\ln 2}{t_{1 / 2}}$ for the half-life. We need to find a similar expression for $t_{1 / 3}$ .

By definition, at $t = t_{1 / 3}$ , $P_{A} = \frac{P_{0}}{3}$ .

Plugging this into the first order equation: $k = \frac{1}{t_{1 / 3}} \ln \frac{P_{0}}{P_{0} / 3} = \frac{1}{t_{1 / 3}} \ln 3$

Therefore, the rate constant is $k = \frac{\ln 3}{t_{1 / 3}}$ .

Step 4: Find the Total Pressure at t = t_1/3

We need to find $P_{t}$ when $t = t_{1 / 3}$ . We know that at this time, $P_{A} = \frac{P_{0}}{3}$ .

From the pressure relation we

Experiment	[A] (inmolL^-1)	[B] (inmolL^-1)	Initilalrate of reaction (in mol L^–1min^–1)
I	0.10	0.20	6.93 × 10^-3
II	0.10	0.25	6.93 × 10^-3
III	0.20	0.30	1.386 × 10^-2

Detailed Solution

Understanding the First Order Reaction and Pressure Changes

Step 1: Relate Concentration to Pressure

Step 2: Apply the First Order Kinetics Equation

Step 3: Find the Expression for the Rate Constant k

Step 4: Find the Total Pressure at t = t1/3

Student who searched for similar questions

Questions 1

Questions 2

Questions 3

Questions 4

Questions 5

Step 4: Find the Total Pressure at t = t_1/3