. The thermal decomposition of dinitrogen pentoxide in the gas phase to give nitrogen dioxide and oxygen is a first order reaction. If the half life at 55 °C is 410 s, what is the value of the rate constant k? If the initial concentration of dinitrogen pentoxide in an expt at 55 °C was 0.200 M what will be the concentration of dinitrogen pentoxide, nitrogen dioxide and oxygen after 4 half-lives?
Let the initial and final concentrations be Co and Ct.
To find k:
ln(Ct/Co)=-kt
ln(0.5) = -k(410s)
k = ____?
To find the concentration at (4)(410s), use the same equation:
ln(Ct/Co)=-kt
Use the value of k from the first part, and t = 4x410=1640s, then solve for for Ct.
To find the value of the rate constant (k) for the thermal decomposition of dinitrogen pentoxide, we can use the half-life (t1/2) and the formula for first-order reactions:
t1/2 = ln(2) / k
Rearranging the formula, we can solve for k:
k = ln(2) / t1/2
Plugging in the given half-life, we can calculate the rate constant:
k = ln(2) / 410 s
Now, let's calculate the value of k using this formula:
k = 0.693 / 410 s
k ≈ 0.00169 s^(-1)
Now, let's move on to the second part of the question. If the initial concentration of dinitrogen pentoxide is 0.200 M and the reaction has a first-order rate constant, we can use the equation for a first-order reaction to determine the concentrations after a certain number of half-lives:
[Reactant] = [Reactant]₀ * 2^(-n)
where [Reactant] is the final concentration, [Reactant]₀ is the initial concentration, and 'n' is the number of half-lives.
Given that we want to find the concentrations after 4 half-lives, we will substitute n = 4 in the equation.
For dinitrogen pentoxide:
[DNPO] = [DNPO]₀ * 2^(-4)
[DNPO] = 0.200 M * 2^(-4)
[DNPO] = 0.200 M * 0.0625
[DNPO] = 0.0125 M
For nitrogen dioxide (NO₂):
[NO₂] = [NO₂]₀ * (1 - 2^(-n))
[NO₂] = 0.200 M * (1 - 2^(-4))
[NO₂] = 0.200 M * (1 - 0.0625)
[NO₂] = 0.200 M * 0.9375
[NO₂] = 0.1875 M
For oxygen (O₂):
[O₂] = [O₂]₀ - [DNPO]
[O₂] = 0.200 M - 0.0125 M
[O₂] = 0.1875 M
After 4 half-lives, the concentration of dinitrogen pentoxide (DNPO) is 0.0125 M, the concentration of nitrogen dioxide (NO₂) is 0.1875 M, and the concentration of oxygen (O₂) is also 0.1875 M.