2/8/2017 11:30 PM A 61.5/100 Gradebook Print Calculator Periodic Table Question

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2/8/2017 11:30 PM A 61.5/100 Gradebook Print Calculator Periodic Table Question 7 of 13 Map a Sapling Learning macmillan learning A reactant decomposes with a half-life of 127 s when its initial concentration is 0.374 M. When the initial concentration is 0.796 M, this same reactant decomposes with the same half-life of 127 s. What is the order of the reaction? O 0 What is the value and unit of the rate constant for this reaction? Number Select answer a Previous ® Give Up & View Solution Check Answer Next Exit B Hint

Explanation / Answer

Part A:

Let us consider that the reaction be of nth with respect to reactant A,

So,
d[A]/dt = - rate = -k[A]

Above k is the rate constant

Now, solve this differential equation with initial condition:
t=0 [A] = [A]
you get the associated integrated rate laws:
for n 1
[A]¹ = [A]¹ + (n - 1)kt
for n=1
ln[A] = ln[A] - kt

By setting
[A] = [A]/2 at t = t½
and resolving for t½ you get the half-life relations:
for n 1
t½ = ( ((1/2)¹ - 1) / (n - 1)k ) [A]¹ = ( 2¹ - 1) / ( (n - 1)k[A]¹ )
for n=1
t½ = ln(2) / k

Again the second formula shows that half-life for a first order reaction does not depend on initial concentration. Apparently this is not for this reaction. So the first formula applies for this reaction.
Be setting
C = ( 2¹ - 1) / ( (n - 1)k)
you get
t½ = C/[A]¹
That means half-life for reaction, which ist not of 1st order, is inversely proportional to the initial concentration raised to the power n-1.

To compute the reaction order consider the quotient of the two half-lifes:
t½, / t½, = ( C/[A], ¹ ) / ( C/[A],¹ ) = ( [A],/[A], )¹
To get n-1 take the logarithm an apply the identity
ln(x) = nln(x)
=> ln(t½, / t½,) = ln( ( [A],/[A], )¹ ) = (n - 1)ln( [A],/[A], )
=> n - 1 = ln(t½, / t½,) / ln( [A],/[A], )

Now, put the values from the given problem -

=> n - 1 = ln(127 s / 127 s) / ln( 0.374 M / 0.796 M ) = 0

=> n = 1

So, the reaction is of first order.

Part B:

t (1/2) = 127 s.

Now, write the formula for first order reaction -

t(1/2) = ln(2) / k

k = ln(2) / t(1/2) = 0.693 / 127 = 0.00546 s^-1 = 5.46 x 10^-3 s^-1.

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