The exponential function

The relationship between three numbers represented as a, b  and c:   

forms the basis of the definition of a logarithm: 

c  = log_b a

Logarithms can be defined to any base b, but commonly to base 10. The second important value that base b can take is the exponential number (also known as Euler's number) which approximates to 2.718, and which is denoted by the symbol e. The exponential of a variable x is then written as e^x, or exp(x) which is particularly useful when x  is replaced by a more complicated expression. (Note: textbooks traditionally introduced the exponential function as a power series,¹ but more recently this approach has remained the province of more advanced textbooks.) 

What is so special about e?

If we were to plot a graph of e^x against x, we obtain a curve. The gradient of the curve constantly changes, but we can determine its value at any point by measuring the gradient of the tangent to the curve at that point. It turns out that for our graph e^x versus x, the gradient at any point is equal to the value of e^x itself. In chemistry many processes can be described mathematically by the exponential function. And, as in the case of logarithms, in chemistry we only take the exponential of a pure number (without units) and the resulting value will also be a pure number. Students do not generally have difficulty in working out simple exponents with a calculator or computer. Nevertheless, when applied to real problems the computation of exponential values requires accurate work involving careful manipulation of expressions and units, and students need to have a thorough understanding of the relationships between decimals, percentage and ratio.² 

Teaching strategies

As teachers we are happy to sketch curves of exponential growth and decay, but once we attempt to add numbers to our curves we obtain some far less familiar shapes.³ Simply plotting various exponential graphs using a spreadsheet program is consequently a useful activity to enable students to gain familiarity with the exponential function. 

Another useful strategy involves first introducing the quantitative concept in context.⁴ Siphons, water clocks, cooling coffee and leaking capacitors have been used as experimental examples to teach the concepts and mathematics of radioactive decay.⁵ Students can then analyse experimental data to determine the nature of the decay process, and make prediction both forwards and backwards in time as would be done for the radioactive process.  

Example 1

Radioactive decay is possibly the best known example of a process which is described by the exponential function. The number of radioactive nuclei N present after time t is given by the equation: 

N  = N₀e ^-λt  (i) 

whereN₀ is the number of radioactive nuclei initially present (ie when t = 0). The constantλ is related to the nuclei's half-life t_½ by the equation:

t_½ = 0.693/λ (ii) 

The half-lifet_½ is the time taken for the number of nuclei present to fall to half of their initial value; this value is constant irrespective of the starting value.

The activity A of a radioactive substance can be shown to be described by the related equation:   

A  = A₀e ^-λt  (iii) 

whereA₀ is the initial activity. 

The isotope ¹⁴ C has a half-life of 5730 years. Starting with equation (ii), multiplying both sides by λ then dividing both sides by t½, gives:

λ = 0.693 / t½  = 0.693 / 5730y = 1.21 × 10^-4 y

The activity A₀ of ¹⁴ C in living organisms is 12.5 min^-1 (g C)^-1. We now have enough information to determine the activity A at any subsequent time. For example, after 10 000 years,  

A  = 12.5 min^-1  (g C)^-1× e ^-1.21× 10^-4 y^-1× 10⁵ y 

= 12.5 min^-1  (g C) × e ^{-1.21? 10}

= 12.5 min^-1  (g C)^-1× e ^-12.1

= 12.5 min^-1  (g C)^-1× 5.56 ? 10^-6

= 6.95 × 10^-5  min^-1  (g C)^-1

Notice how the units cancel so that we are taking the exponential of a pure number. Note also the use of the symbol y to represent the unit of year. There is in fact no accepted convention for this because it is not an SI unit; yr, yrs or even a (for the Latin annus) could have been used. 

Example 2

The exponential function also allows us to determine the relative populations of two quantum states 1 and 2. If they differ in energy by ΔE, the ratio of the populations n₁ and n₂ is given by:

n₂/n₁ = e ^-ΔE/kT  (iv) 

where k  is the Boltzmann constant (1.381 × 10^-23 J K^-1) and T is the absolute temperature. We will assume that state 2 has higher energy than state 1. 

The excited state of molecular oxygen dissociates into two oxygen atoms, one of which is an energy level 190 kJ mol^-1 above the ground state. To use equation (iv), we need to convert the given value of ΔE  to that for atoms; we do this by dividing by the Avogadro constant to give: 

ΔE  = 190 × 10³ J mol^-1 / 6.022 × 10²³ J K^-1

= 3.16 × 10^-19 J 

The ratio of the number of molecules in the two states at room temperature is then given by: 

n₂/n₁ = e ^{-3.16 × 10} -19J / 1.381 × 10^-23 J K^-1× 298 K

= e^-76.8

= 4.42 × 10^-34

which is negligible. Repeating the calculation at a temperature of 10,000K gives n₂/n₁ = 0.10, so we see that the higher energy state is sparsely populated even at higher temperatures. 

Once again in this calculation we see that units within the exponential cancel, and this check should always be made when doing such calculations.

Related Links

Victoria Department of Education
Mathematics developmental continuum

Teaching Quantitative skills in Geosciencs
Teaching the exponential function

Application of the exponential
Application of the exponential