Solution:
Step 1: Assume that the mass of a compound is 100 g.
[tex]\text{mass C = 74.02 g}[/tex]
[tex]\text{mass H = 8.710 g}[/tex]
[tex]\text{mass N = 17.27 g}[/tex]
Step 2: Calculate the number of moles of each element.
[tex]n \: \text{C = 74.02 g} × \frac{\text{1 mol}}{\text{12.01 g}} = \text{6.163 mol}[/tex]
[tex]n \: \text{H = 8.710 g} × \frac{\text{1 mol}}{\text{1.008 g}} = \text{8.641 mol}[/tex]
[tex]n \: \text{N = 17.27 g} × \frac{\text{1 mol}}{\text{14.01 g}} = \text{1.233 mol}[/tex]
Step 3: Represent an empirical formula.
[tex]\text{empirical formula} = \text{C}_{x}\text{H}_{y}\text{N}_{z}[/tex]
Step 4: Divide the number of moles of each element by the least number of moles.
[tex]x = \frac{\text{6.163 mol}}{\text{1.233 mol}} = 5[/tex]
[tex]y = \frac{\text{8.641 mol}}{\text{1.233 mol}} = 7[/tex]
[tex]z = \frac{\text{1.233 mol}}{\text{1.233 mol}} = 1[/tex]
Step 5: Write the empirical formula.
[tex]\text{empirical formula} = \text{C}_{5}\text{H}_{7}\text{N}[/tex]
Step 6: Represent a molecular formula.
[tex]\text{molecular formula} = (\text{C}_{5}\text{H}_{7}\text{N})_{n}[/tex]
Step 7: Calculate the empirical weight.
empirical weight = (12.01 g/mol × 5) + (1.008 g/mol × 7) + (14.01 g/mol × 1)
empirical weight = 81.12 g/mol
Step 8: Calculate the molecular weight.
[tex]\text{molecular weight} = \frac{\text{mass of compound}}{\text{moles of compound}}[/tex]
[tex]\text{molecular weight} = \frac{\text{40.57 g}}{\text{0.2500 mol}}[/tex]
[tex]\text{molecular weight = 162.3 g/mol}[/tex]
Step 9: Divide the molecular weight by the empirical weight.
[tex]n = \frac{\text{molecular weight}}{\text{empirical weight}}[/tex]
[tex]n = \frac{\text{162.3 g/mol}}{\text{81.12 g/mol}}[/tex]
[tex]n = 2[/tex]
Step 10: Multiply the subscripts by the value of n to obtain the molecular formula.
[tex]\text{molecular formula} = (\text{C}_{5}\text{H}_{7}\text{N})_{2}[/tex]
[tex]\boxed{\text{molecular formula} = \text{C}_{10}\text{H}_{14}\text{N}_{2}}[/tex]
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