Distributions
Temperature is a measure of the average kinetic energy of
the particles in a sample, but, how are the particles distributed in terms of energy?
Bell Curve – Maxwell-Boltzmann Distribution
Statistical curves like this are typically obtained when:
(a)
there are repeated scientific readings and/or
(b)
measuring living characteristics (height, weight, IQ, etc)
The graph always shows "number of readings" on the y-axis and "value of readings" on the x-axis. The curve is called a "bell curve" due to its shape. The "0", marking the highest point on the curve, indicates the average reading. The "σ" represents the standard deviation (how far from the average is a reading).
For example:
Imagine we took height measurements from everyone at school. We might find that the average height is 5'6". Obviously, not everyone has a height of 5'6", some people are shorter and some people are taller. We might arrive at the following: 5'6" ∓ 3". The "∓ 3" is the standard deviation. Thus, using statistically defined brackets:
-
68% of readings are between σ
to -σ (68% of people are between 5'3 and 5'9")
-
95% of readings are between
2σ to -2σ (95% of people are between 5'0" and 6'0")
-
100% of readings are between 3σ
to -3σ (100% of people are between 4'9" and 6'3")
So, looking at the above curve, the majority of people are of average height. The number of people who are taller or shorter decreases as you move to either side of the average.
Statistics Terminology
mode – most frequently occurring reading
mean – arithmetical average
median – middle value in a group of readings
Kinetic Energy Distribution Curves - Theoretical Effect of
Temperature
We use the graph above to show how EK is distributed over a sample of particles at a given temperature (note: T2 > T1).
The sample at the higher temperature (T2) has more particles with the minimum kinetic energy (the area under the bell curve to the right of the line), therefore the reaction will proceed faster at T2 than at T1.
Recall that this minimum kinetic energy is called the activation energy (Ea) or threshold energy – for most reactions, the activation energy is quite high and at lower temperatures, very few particles have enough energy.
Theoretical Effect of Chemical Nature of Reactant
When Cu is added to HCl(aq) , it reacts slowly (it has the higher activation energy on the above graph and thus, has few particles with the necessary energy to react.
When Ca is added to HCl(aq), it reacts very quickly (it has the lower activation energy on the above graph and thus, has many particles with the necessary energy to react).
To explain why some substances are more reactive, we have to consider many variables, such as, the atomic structure of the reactants, the nature of the bonds and the type of reaction occurring.
As discussed earlier, an effective collision requires a minimum activation (or
threshold) energy.
There are two ways that the chemical nature of the reactants
can affect the activation energy:
(a)
some molecules have relatively weak bonds, so the threshold
energy is low, which results in a large fraction of the molecules having the
ability to react
(b)
reaction geometry (reaction sterics) - the orientation of a
molecule is important in many collisions - although there may be many effective
collisions occurring between reactants, a reaction will not occur if the
reactants are not aligned properly
Theoretical Effect of Concentration & Surface Area
When Mg is added to concentrated HCl(aq), it reacts more quickly
than when added to dilute HCl(aq). If the concentration is doubled, there are twice as many particles available to collide effectively, thus the rate should double.
When the Mg is cut into small pieces, it reacts more quickly
than when it is in a large lump. If the substance is divided up, there is more surface area
available for reaction, however, a solid can only be divided up so much.
Theoretical Effect of Catalysis
A catalyst lowers the activation energy by providing a
different reaction profile. This can be illustrated using a potential energy diagram (notice that the top of the curve is lowered, indicating that less energy is required for reactants to become products).
Or we can look at the
bell curve (notice that the activation energy line moves to the left,
resulting in more particles having the minimum energy to react).
Catalysts come in two varieties:
- homogeneous catalyst - found in the same phase as the
reactants (for instance, H+(aq) in solution reaction).
- heterogeneous catalyst - exists in different phase than
reactants (for instance, Pt(s) in a gaseous reaction).
Enzymes are biological catalysts.
The opposite of a catalyst is an inhibitor, which would increase the activation energy and thus, slow the reaction rate.
