The coefficient A in the Arrhenius equation is the frequency factor. What are the units of A for a first-order reaction? Justify your answer mathematically.
The units of A in the Arrhenius equation for a first-order reaction are s-1. This can be justified mathematically by noting that the Arrhenius equation for a first-order reaction is:
k = A e^(-Ea/RT)
where k is the rate constant, A is the frequency factor, Ea is the activation energy, R is the gas constant, and T is the temperature.
Since the rate constant has units of s-1, the units of A must also be s-1 in order for the equation to be dimensionally consistent.
To determine the units of coefficient A in the Arrhenius equation for a first-order reaction, we first need to understand the equation itself.
The Arrhenius equation is given as follows:
k = A * e^(-Ea/RT)
Where:
- k is the rate constant of the reaction
- A is the frequency factor (pre-exponential factor)
- Ea is the activation energy of the reaction
- R is the gas constant
- T is the temperature in Kelvin
For a first-order reaction, the rate equation is:
rate = k * [A]
Where:
- rate is the rate of the reaction
- [A] is the concentration of reactant A
Since the rate constant k is the proportionality constant that relates the rate of the reaction to the concentrations of the reactants, we can determine its units from the units of rate and concentration.
For a first-order reaction, the rate is expressed in units of mol L^-1 s^-1, and the concentration is expressed in units of mol L^-1.
So, substituting these units into the rate equation for a first-order reaction, we have:
mol L^-1 s^-1 = (A) * mol L^-1
To cancel out the units of concentration mol L^-1 on both sides of the equation, we can divide both sides by mol L^-1, resulting in:
s^-1 = (A)
Thus, we can conclude that the units of coefficient A for a first-order reaction are inverse seconds (s^-1).
The Arrhenius equation relates the rate constant (k) of a reaction to temperature (T) and activation energy (Ea). It is given by:
k = A * e^(-Ea/RT)
Where:
k = rate constant
A = frequency factor or pre-exponential factor
e = the mathematical constant approximately equal to 2.71828
Ea = activation energy
R = gas constant
T = temperature
For a first-order reaction, the rate constant (k) can be expressed as:
k = ln(2) / t1/2
Where:
ln(2) is the natural logarithm of 2
t1/2 is the half-life of the reaction
To determine the units of A for a first-order reaction, we can equate the two expressions for k:
k = A * e^(-Ea/RT) = ln(2) / t1/2
Rearranging this equation, we can solve for A:
A = (ln(2) / t1/2) / e^(-Ea/RT)
Now let's analyze the units of each term. In the numerator, we have ln(2) / t1/2, which has the units of s^-1 (inverse seconds) since t1/2 is the time taken for the concentration of the reactant to halve.
In the denominator, we have e^(-Ea/RT), which is a dimensionless quantity since Ea (J/mol), R (J/mol·K), and T (K) all have consistent units.
Therefore, the overall units of A for a first-order reaction are s^-1.
The frequency factor (A) represents the rate of successful molecular collisions per unit time. In the context of first-order reactions, it indicates the rate at which reactant molecules collide with enough energy and proper orientation to undergo the reaction. The units of A are chosen such that when multiplied by the exponential term, it yields a rate constant with the appropriate units for the reaction order.