Example 3
The next group of hydrocarbons in organic chemistry is called the alkenes. This group contains a double bond between at least two of the carbon atoms. The first three simple alkenes are ethene, propene, and butene. The number of carbon and hydrogen atoms in each molecule is identified below:

Alkene  Ethene  Propene  Butene
 Number of Carbon Atoms
\(2\) \(3\) \(4\)
 Number of Hydrogen Atoms
\(4\) \(6\) \(8\)

Using the table,

  1. Determine the general formula relating the number of hydrogen atoms to the number of carbon atoms for alkenes.  Remember to define the variables.

    Step 1:  Write down what is given.

    Let \(t_n \) represent the number of hydrogen atoms, and let \(n \) represent the number of carbon atoms. The general term is:

    \(t_n = t_1 + ( n - 1)d \)

    Looking at the table, the information you are given is:

    \( \begin{align}
     t_2&= 4 \\
     d&= 6 - 4 = 2 \\
     \end{align} \)

    Notice that you do not know the first term because there is not an alkene with only one carbon.

    Step 2: Determine the first term, \(t_1\), using \(t_2 = 4\).

    \( \begin{align}
     t_n &= t_1 + \left( {n - 1} \right)d \\
     4 &= t_1 + \left( {2 - 1} \right)2 \\
     4 &= t_1 + 2 \\
     2 &= t_1  \\
     \end{align} \)

    Step 3: Produce a formula.

    \( \begin{align}
     t_n &= t_1 + \left( {n - 1} \right)d \\
     t_n &= 2 + \left( {n - 1} \right)2 \\
     t_n &= 2 + 2n - 2 \\
     t_n &= 2n \\
     \end{align} \)
  2. \(3\), \(6\) dimethyl-\(2\)-octene is an alkene with \(20\) hydrogen atoms. Determine the number of carbon atoms in the molecule.

    Using the formula from part a., substitute \(20\) for \(t_n \) , and solve for \(n \).

    \( \begin{align}
     t_n&= 2n \\
     20&= 2n \\
     10&= n \\
     \end{align} \)

    There are \(10\) carbon atoms in the molecule.
As seen in Example 3, the general term can be used to determine unknown values for \(n \), \(t_1 \), and \(d \), in addition to \(t_n \).