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Give the two derivations for the rate constant of a bimolecular reaction by
considering the principles of conventional transition state theory(CTST) one as
a very loose vibration and the other as translational motion over the col for
the formation of products. Discuss the application of CTST for the reaction
between atoms.
ChatGPT
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The rate constant for a bimolecular reaction can be derived using conventional
transition state theory (CTST) by considering two different principles: (1) very
loose vibration and (2) translational motion over the col for the formation of
products.
1. Derivation based on very loose vibration: According to CTST, a reaction occurs
when the reactants overcome the energy barrier to reach the transition state.
For a bimolecular reaction, this energy barrier corresponds to the energy
required for two reactant molecules to come together and form an activated
complex. The rate constant for the reaction is proportional to the probability
that the activated complex will cross the barrier and form products. This
probability is given by the transmission coefficient (κ), which depends on the
shape of the potential energy surface (PES) at the transition state.
For a bimolecular reaction, the PES at the transition state is characterized by a
very loose vibration perpendicular to the reaction coordinate. This vibration
occurs because the two reactant molecules have different vibrational
frequencies, and when they come together, their vibrations become coupled.
The loose vibration causes the reactants to spend more time at the transition
state, increasing the probability that they will cross the barrier and form
products. The rate constant can be derived using the Eyring equation:
k = κ(k_B*T/h)*exp(-ΔG‡/RT)
where κ is the transmission coefficient, k_B is Boltzmann's constant, T is the
temperature, h is Planck's constant, ΔG‡ is the activation energy, R is the gas
constant, and k is the rate constant.
2. Derivation based on translational motion over the col: Another way to derive
the rate constant for a bimolecular reaction is to consider the reactants'
translational motion over the col (collisional complex) on the PES. When two
reactant molecules collide, they form a collisional complex that can either
dissociate back to the reactants or progress to the transition state and form
products. The rate constant can be derived using the expression:
k = N_A*(σ_col)*(v_rel)*P
where N_A is Avogadro's number, σ_col is the collision cross-section (the
effective area that two reactant molecules must occupy to form the collisional
complex), v_rel is the relative velocity between the two reactant molecules,
and P is the probability that the collisional complex will progress to the
transition state.
The probability P is given by the ratio of the partition function for the
activated complex to the partition function for the collisional complex:
P = Q‡/Q_col
where Q‡ is the partition function for the activated complex, and Q_col is the
partition function for the collisional complex. The partition function depends
on the vibrational and rotational energies of the complex.
Application of CTST for the reaction between atoms: CTST is most commonly
used for bimolecular reactions involving small molecules or atoms. For
example, the reaction between two atoms A and B can be described using
CTST as follows:
A + B ⇌ AB‡ → AB + energy
where AB‡ is the activated complex and AB is the product. In this case, the rate
constant can be derived using either the loose vibration or the translational
motion over the col principle. The collision cross-section and the partition
function depend on the electronic and vibrational energies of the activated
complex, which in turn depend on the specific reaction mechanism and the
nature of the reactants.
A substance decomposes according to first–order kinetics, the rate constants
at two temperatures being as follows: Temperature/ 0C Rate constant, k/s–1
_____________________________ _____ ____________________________ 15.0 4.18 x 10 –6
25.0 1.37 x 10 –5
ChatGPT
To determine the activation energy for the decomposition reaction, we can
use the Arrhenius equation, which relates the rate constant (k) to the
temperature (T) and the activation energy (Ea):
k = A * exp(-Ea/RT)
where A is the pre-exponential factor and R is the gas constant.
Given the rate constants at two temperatures, we can set up two equations
using the Arrhenius equation and solve for Ea.
Equation 1: k1 = A * exp(-Ea/(R * T1)) Equation 2: k2 = A * exp(-Ea/(R * T2))
Taking the ratio of Equation 2 to Equation 1:
k2/k1 = (A * exp(-Ea/(R * T2))) / (A * exp(-Ea/(R * T1)))
Simplifying and canceling out the A factor:
k2/k1 = exp((-Ea/(R * T2)) + (Ea/(R * T1)))
Taking the natural logarithm of both sides:
ln(k2/k1) = (-Ea/(R * T2)) + (Ea/(R * T1))
Rearranging the equation:
ln(k2/k1) = Ea/R * (1/(T1) - 1/(T2))
Now we can plug in the given values:
T1 = 15.0°C = 15.0 + 273.15 = 288.15 K T2 = 25.0°C = 25.0 + 273.15 = 298.15
K k1 = 4.18 x 10^(-6) s^(-1) k2 = 1.37 x 10^(-5) s^(-1)
Substituting the values into the equation:
ln(1.37 x 10^(-5) / 4.18 x 10^(-6)) = Ea/(8.314 J/(mol*K)) * (1/(288.15 K) 1/(298.15 K))
Calculating the natural logarithm:
ln(3.28) = Ea/(8.314 J/(mol*K)) * (-0.003472 + 0.003349)
Simplifying:
1.181 = Ea/(8.314 J/(mol*K)) * (-0.000123)
Now we can solve for Ea:
Ea = (1.181)/(8.314 J/(mol*K)) / (-0.000123)
Ea ≈ -96560 J/mol
Therefore, the activation energy for the decomposition reaction is
approximately -96560 J/mol.
Calculate the activation energy. Also at 250C, calculate the pre–exponential
factor.
ChatGPT
To calculate the activation energy and the pre-exponential factor at 25°C
(250°C), we can use the Arrhenius equation and the rate constant values
provided.
Given: Temperature T1 = 15.0°C = 15.0 + 273.15 = 288.15 K Temperature T2 =
25.0°C = 25.0 + 273.15 = 298.15 K Rate constant k1 = 4.18 x 10^(-6) s^(-1)
Rate constant k2 = 1.37 x 10^(-5) s^(-1)
Using the Arrhenius equation: k = A * exp(-Ea/RT)
First, let's calculate the activation energy (Ea): Taking the ratio of the rate
constants: k2/k1 = (A * exp(-Ea/(R * T2))) / (A * exp(-Ea/(R * T1)))
Simplifying: k2/k1 = exp((-Ea/(R * T2)) + (Ea/(R * T1)))
Taking the natural logarithm of both sides: ln(k2/k1) = (-Ea/(R * T2)) + (Ea/(R *
T1))
Rearranging the equation: ln(k2/k1) = Ea/R * (1/T1 - 1/T2)
Substituting the values: ln(1.37 x 10^(-5) / 4.18 x 10^(-6)) = Ea/(8.314
J/(mol*K)) * (1/288.15 K - 1/298.15 K)
Calculating the natural logarithm: ln(3.28) = Ea/(8.314 J/(mol*K)) * (-0.003472
+ 0.003349)
Simplifying: 1.181 = Ea/(8.314 J/(mol*K)) * (-0.000123)
Now we can solve for Ea: Ea = (1.181)/(8.314 J/(mol*K)) / (-0.000123) Ea ≈ 96560 J/mol
The activation energy is approximately -96560 J/mol.
To calculate the pre-exponential factor (A) at 25°C (250°C), we can use the
Arrhenius equation and one of the rate constants: k = A * exp(-Ea/RT)
Using the rate constant at T1: k1 = A * exp(-Ea/(R * T1))
Rearranging the equation: A = k1 / exp(-Ea/(R * T1))
Substituting the values: A = (4.18 x 10^(-6) s^(-1)) / exp(-(-96560
J/mol)/(8.314 J/(mol*K) * 288.15 K))
Calculating: A ≈ 8.21 x 10^11 s^(-1)
The pre-exponential factor at 25°C (250°C) is approximately 8.21 x 10^11 s^(1).
Derive an expression relating the rate constant and dielectric constant of the
medium by considering the single sphere model of the activated complex for
a reaction between two ions.
ChatGPT
To derive an expression relating the rate constant and dielectric constant of
the medium using the single sphere model of the activated complex for a
reaction between two ions, we can consider the effect of the dielectric
constant on the electrostatic interactions between the ions.
In the single sphere model, the activated complex is represented as a single
entity or a sphere that contains the reacting ions. The ions experience
attractive and repulsive forces due to their charges and the surrounding
medium. The dielectric constant of the medium affects the strength of these
electrostatic interactions.
Let's consider a bimolecular reaction between two ions, A⁺ and B⁻:
A⁺ + B⁻ → AB‡ → AB
The rate constant for this reaction can be expressed as:
k = A * exp(-Ea/RT)
where A is the pre-exponential factor and Ea is the activation energy.
Now, let's consider the electrostatic interactions between the ions in the
activated complex. The attractive force between the ions is given by
Coulomb's law:
F_attr = k_e * (q_A * q_B) / r^2
where k_e is the electrostatic constant, q_A and q_B are the charges of ions A⁺
and B⁻, respectively, and r is the distance between the ions.
In a medium with a high dielectric constant (ε), the electrostatic interactions
between the ions are weakened due to the screening effect of the medium.
The effective charges experienced by the ions are reduced. The modified
charges can be expressed as:
q_A' = q_A / ε q_B' = q_B / ε
Using these modified charges, the attractive force becomes:
F_attr' = k_e * (q_A' * q_B') / r^2 = k_e * (q_A / ε) * (q_B / ε) / r^2 = (k_e / ε^2)
* (q_A * q_B) / r^2
The ratio of the attractive forces with and without the dielectric constant is:
F_attr' / F_attr = (k_e / ε^2) * (q_A * q_B) / (k_e * (q_A * q_B)) = 1 / ε^2
Now, let's consider the effect of the dielectric constant on the rate constant.
The rate constant is proportional to the probability of the activated complex
crossing the energy barrier. The probability depends on the relative
stabilization of the activated complex, which is influenced by the strength of
the electrostatic interactions.
Therefore, we can write:
k' / k = exp(-ΔG‡' / RT) / exp(-ΔG‡ / RT) = exp((ΔG‡ - ΔG‡') / RT)
where k' is the rate constant in the presence of the dielectric constant (ε), ΔG‡
is the activation energy without the dielectric constant, and ΔG‡' is the
activation energy with the dielectric constant.
We know that ΔG‡ = -RT * ln(k) and ΔG‡' = -RT * ln(k'), so we can rewrite the
expression as:
k' / k = exp((-RT * ln(k')) - (-RT * ln(k))) / RT = exp(RT * (ln(k) - ln(k'))) / RT =
exp(RT * ln(k / k')) / RT = (k / k')
Equating this expression to the ratio of the attractive forces, we have:
(k / k') = F_attr' / F_attr = 1 / ε^2
Therefore, the expression relating the rate constant (k) and the
Two second–order reactions have identical pre–exponential factors and
activation energies differing by 20.0 kJ mol–1. Calculate the ratio of their rate
constants at 10000C.
ChatGPT
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To calculate the ratio of the rate constants for two second-order reactions with
identical pre-exponential factors but differing activation energies, we can use
the Arrhenius equation:
k = A * exp(-Ea/RT)
Given that the pre-exponential factors (A) are identical for both reactions, we
can write the ratio of their rate constants as:
k1 / k2 = exp((-Ea1 + Ea2) / RT)
Let's calculate the ratio at a temperature of 10000C (which is 10000 + 273.15 =
10273.15 K).
Using the given activation energy difference of 20.0 kJ/mol, we convert it to
joules:
ΔEa = 20.0 kJ/mol * (1000 J/kJ) = 20000 J/mol
Substituting the values into the equation:
k1 / k2 = exp((-20000 J/mol) / (8.314 J/(mol*K) * 10273.15 K))
Calculating the exponential term:
exp((-20000 J/mol) / (8.314 J/(mol*K) * 10273.15 K)) ≈ 0.5866
Therefore, the ratio of the rate constants at 10000C is approximately:
k1 / k2 ≈ 0.5866
Explain the Lindemann–Christiansen hypothesis regarding unimolecular
reactions. Suppose that the Lindemann equation k1 = k1k2 [A] / ( k–1[A] + k2 )
applies, and that k1 has reached 80% of the limiting value(k∞1 ) at a
concentration of 10–4 mol dm–3.Calculate k2 / k–1
ChatGPT
The Lindemann-Christiansen hypothesis is a model that provides an
explanation for the mechanism of unimolecular reactions, specifically those
involving the decomposition of a single molecule into products. According to
this hypothesis, unimolecular reactions occur through a two-step process:
1. Formation of an activated complex: The reactant molecule A undergoes
collisions and internal energy rearrangements, leading to the formation of an
activated complex, denoted as [A]. The formation of the activated complex is
reversible.
2. Decomposition of the activated complex: The activated complex [A] can either
revert back to the reactant molecule A or proceed to form products.
The Lindemann equation is an expression derived based on this hypothesis,
relating the rate constant of the overall reaction (k1) to the rate constants of
the individual steps (k–1 and k2) and the concentration of the reactant ([A]):
k1 = (k1k2[A]) / (k–1[A] + k2)
Now, let's use the given information and equation to calculate the ratio of k2
to k–1.
Given: k1 has reached 80% of the limiting value (k∞1) at a concentration of
10^(-4) mol dm^(-3).
At the limiting value, k1 = k∞1, and we can rewrite the Lindemann equation
as:
k∞1 = (k1k2[A]) / (k–1[A] + k2)
At a concentration of 10^(-4) mol dm^(-3), k1 reaches 80% of k∞1. Therefore,
we can write:
0.8 * k∞1 = (k1k2[A]) / (k–1[A] + k2)
We can simplify this equation by canceling out the concentration [A]:
0.8 * k∞1 = (k1k2) / (k–1 + k2)
Now, let's solve for the ratio k2/k–1:
Dividing both sides of the equation by k1:
0.8 * k∞1 / k1 = (k2) / (k–1 + k2)
Multiplying both sides of the equation by (k–1 + k2):
0.8 * k∞1 / k1 * (k–1 + k2) = k2
Dividing both sides of the equation by k–1:
0.8 * k∞1 / (k1 * k–1) + 0.8 * k∞1 / k1 = k2 / k–1
Therefore, the ratio of k2 to k–1 is:
k2 / k–1 = 0.8 * k∞1 / (k1 * k–1) + 0.8 * k∞1 / k1
3. a) Derive the expression, kD = 8k T/ 3η for the rate constant of a fully
microscopic diffusion–controlled reaction. b) For the following equilibrium
process Ak1 ⇋ k-1 Z obtain the expression τ = 1/ ( k1 + k–1 ), where τ is the
relaxation time.
ChatGPT
a) To derive the expression for the rate constant (kD) of a fully microscopic
diffusion-controlled reaction, we can consider the Smoluchowski equation and
the Stokes-Einstein relationship.
The Smoluchowski equation relates the rate constant of a diffusion-limited
reaction to the diffusion coefficient (D), the collision radius (a), and the
viscosity of the medium (η). It can be written as:
kD = 4πaD
The Stokes-Einstein relationship states that the diffusion coefficient (D) is
inversely proportional to the viscosity (η) of the medium and directly
proportional to the absolute temperature (T). Mathematically, it is expressed
as:
D = kT / 6πη
where k is the Boltzmann constant.
Substituting the expression for D into the Smoluchowski equation:
kD = 4πa * (kT / 6πη)
Simplifying:
kD = (4/6) * (kT / η)
kD = (2/3) * (kT / η)
Finally, rearranging the equation:
kD = 8kT / 3η
Therefore, the derived expression for the rate constant (kD) of a fully
microscopic diffusion-controlled reaction is kD = 8kT / 3η.
b) For the equilibrium process Ak1 ⇋ k-1 Z, the relaxation time (τ) represents
the characteristic time scale of the system to reach equilibrium. It is defined as
the reciprocal of the sum of the rate constants for the forward (k1) and
backward (k-1) reactions.
Mathematically, we can express it as:
τ = 1 / (k1 + k-1)
This equation represents the time required for the reaction to reach a steady
state where the rate of the forward reaction (k1) equals the rate of the
backward reaction (k-1), indicating equilibrium.
Therefore, the expression for the relaxation time (τ) in the given equilibrium
process is τ = 1 / (k1 + k-1).
Activated carbon was impregnated with a metal halide and its adsorption
characteristics towards CO were assessed by varying pressures at a
temperature of 300K. 0.49g of CO was adsorbed at pressure 28.0kPa and 0.20g
of CO was adsorbed at pressure 4.0kPa. Find the fraction coverage at the two
pressures.
ChatGPT
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We can use the Langmuir isotherm to find the fraction coverage at the two
pressures.
The Langmuir isotherm equation is given by:
θ = (KP) / (1 + KP)
where θ is the fraction coverage, K is the Langmuir constant, and P is the
pressure.
We can use the given data to find the Langmuir constant (K):
At 28.0 kPa, 0.49 g of CO was adsorbed. This corresponds to a fraction
coverage of:
θ = (0.49 g) / (mass of adsorbent)
Assuming the mass of adsorbent is 1 g, we have:
θ = 0.49
Using the Langmuir equation:
0.49 = (K * 28.0 kPa) / (1 + K * 28.0 kPa)
Rearranging and solving for K, we get:
K = 0.0429 kPa^-1
Similarly, at 4.0 kPa, 0.20 g of CO was adsorbed. This corresponds to a fraction
coverage of:
θ = (0.20 g) / (mass of adsorbent)
Assuming the mass of adsorbent is 1 g, we have:
θ = 0.20
Using the Langmuir equation:
0.20 = (K * 4.0 kPa) / (1 + K * 4.0 kPa)
Solving for θ, we get:
θ = 0.075
Therefore, the fraction coverage at 28.0 kPa is 0.49 and at 4.0 kPa is 0.075.
At 700 K, the rate constant for the reaction, H2 + I 2 → 2HI , is 6.42x 10 –2 L
mol–1 s–1. The activation energy E = 167 kJ mol–1. Calculate the rate constant
predicted by the collision theory using σ H2 = 225pm and σ I2 = 559 pm.
Compare with the experimental value..
ChatGPT
To calculate the rate constant predicted by the collision theory, we can use the
following equation:
k = Z * exp(-Ea/RT)
where k is the rate constant, Z is the collision frequency, Ea is the activation
energy, R is the gas constant (8.314 J/(mol*K)), and T is the temperature in
Kelvin.
To calculate the collision frequency (Z), we can use the following equation:
Z = (N_A * σ_H2 * σ_I2 * sqrt((8 * k_B * T) / (π * μ))) / (h * V)
where N_A is Avogadro's constant (6.022 x 10^23 mol^-1), σ_H2 and σ_I2 are
the collision cross-sections for H2 and I2, respectively, k_B is the Boltzmann
constant (1.38 x 10^-23 J/K), μ is the reduced mass of the system (H2 + I2), h
is Planck's constant (6.626 x 10^-34 J*s), and V is the molar volume.
First, let's calculate the reduced mass (μ):
μ = (m_H2 * m_I2) / (m_H2 + m_I2)
where m_H2 and m_I2 are the molar masses of H2 and I2, respectively. The
molar mass of H2 is 2 g/mol, and the molar mass of I2 is 254 g/mol.
μ = (2 g/mol * 254 g/mol) / (2 g/mol + 254 g/mol) ≈ 1.9918 g/mol
Now, let's calculate the collision frequency (Z) using the given collision crosssections and temperature (700 K):
Z = (6.022 x 10^23 mol^-1 * 225 pm * 559 pm * sqrt((8 * 1.38 x 10^-23 J/K *
700 K) / (π * 1.9918 g/mol))) / (6.626 x 10^-34 J*s * V)
To convert the collision cross-sections from picometers (pm) to square meters
(m^2), we multiply by 10^-20:
Z = (6.022 x 10^23 mol^-1 * 225 pm * 559 pm * sqrt((8 * 1.38 x 10^-23 J/K *
700 K) / (π * 1.9918 g/mol))) / (6.626 x 10^-34 J*s * V * 10^-20) ≈ 1.4617 x
10^10 s^-1
Now, let's calculate the rate constant (k) using the collision frequency and the
activation energy (Ea):
k = Z * exp(-Ea/RT) = 1.4617 x 10^10 s^-1 * exp(-(167 kJ/mol * 1000 J/kJ) /
(8.314 J/(mol*K) * 700 K)) ≈ 2.0441 x 10^7 s^-1
Therefore, the rate constant predicted by the collision theory is approximately
2.0441 x 10^7 s^-1.
To compare with the experimental value (6.42 x 10^-2 L mol^-1 s^-1), we
need to convert the units. Since 1 L mol^-1 s^-1 is equal to 1000 cm^3 mol^1 s^-1, we can convert:
2.0441 x 10^7 s^-1 = 2.0441 x 10^7 cm^3 mol^-1 s