Question

Be sure to answer all parts. The thermal decomposition of phosphine (PH3) into phosphorus and molecular hydrogen is a first-order reaction: 4PH3(g) → P4(g) + 6H2(g) The half-life of the reaction is 35.0 s at 680°C. Calculate the first-order rate constant for the reaction: s−1 Calculate the time required for 77.0 percent of the phosphine to decompose: s

Answer 1

Answer:

k = 0.0198 s⁻¹

t = 74.25 seconds

Explanation:

Given that:

Half life = 35.0 s

$t_{1/2}=\frac {ln\ 2}{k}$

Where, k is rate constant

So,  

$k=\frac {ln\ 2}{t_{1/2}}$

$k=\frac {ln\ 2}{35.0}\ s^{-1}$

The rate constant, k = 0.0198 s⁻¹

Using integrated rate law for first order kinetics as:

$[A_t]=[A_0]e^{-kt}$

Where,  

$[A_t]$ is the concentration at time t

$[A_0]$ is the initial concentration

Given:

77.0 % is decomposed which means that 0.77 of $[A_0]$ is decomposed. So,

$\frac {[A_t]}{[A_0]}$ = 1 - 0.77 = 0.23

t = ?

$\frac {[A_t]}{[A_0]}=e^{-k\times t}$

$0.23=e^{-0.0198\times t}$

t = 74.25 seconds

Answer 2

When in a reaction heat is a reactant then the reaction is called thermal decomposition reaction. The first-order rate constant is 0.0198 per second and the time required is 74.25 seconds.

What is a rate constant?

The rate constant is the specific rate that is the proportionality constant in a reaction and depicts the relation between the chemical reaction rate and the concentration of the reactants.

Rate constant can be given as,

$\rm t\dfrac{1}{2} = \dfrac{ln 2}{k}$

The half-life of the reaction is 35 seconds.

Substituting values in the above equation:

$\begin{aligned}\rm k &= \rm \dfrac{ln\;2}{t\frac{1}{2}}\\\\&= \dfrac{\rm ln\2}{35}\\\\&= 0.0198 \;\rm s^{-1}\end{aligned}$

The time taken can be calculated by the rate law for first order:

$\rm [A_{t}] = [A_{o}]e^{-kt}$

Here,

• Concentration at time t = $\rm [A_{t}]$
• The initial concentration = $\rm [A_{o}]$

Solving for time (t):

$\begin{aligned}\rm \dfrac {[A_{t}]}{[A_{o}]} &= 1 - 0.77\\\\0.23 &= \rm e ^{-0.0198 \times t}\\\\\rm t &= 74.25\;\rm seconds\end{aligned}$

Therefore, the rate constant is 0.0198 per second and the time taken is 74.25 seconds.

Learn more about rate constant here:

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