Question

what is the empirical formula for methamphetamine (shown below)?

Explanation:

Step1: Count atoms in structure

The structure has a benzene ring (6 C), a -CH₂- (1 C), a -CH- (1 C), a -NH- (1 N, 1 H), and two -CH₃-? Wait, no, looking at the structure: benzene ring (C₆H₅), then -CH₂-CH(CH₃)-NH-CH₃? Wait, no, the structure is: benzene attached to -CH₂-CH(NHCH₃)-? Wait, no, the drawing: benzene ring, then a -CH₂- (so C₇), then a -CH- (C₈), then attached to N with a -CH₃ (so N, H, and CH₃: N, H, C). Wait, let's count properly:

Benzene ring: 6 C, 5 H (since it's a phenyl group, C₆H₅). Then the chain: -CH₂- (C₇, H₂), -CH- (C₈, H), then N with H and a -CH₃ (N, H, C₉, H₃). Wait, no, the structure is:

Phenyl (C₆H₅) - CH₂ - CH(NHCH₃). Wait, the N is attached to the CH and to a CH₃. So let's list all atoms:

Carbon: benzene (6) + CH₂ (1) + CH (1) + CH₃ (1) = 6+1+1+1=9? Wait, no, the CH is attached to N and to CH₂, and N is attached to CH₃. So:

C: 6 (benzene) + 1 (CH₂) + 1 (CH) + 1 (CH₃) = 9

H: benzene (5) + CH₂ (2) + CH (1) + NH (1) + CH₃ (3) = 5+2+1+1+3=12? Wait, no: benzene is C₆H₅ (5 H), CH₂ has 2 H, CH has 1 H, NH has 1 H, CH₃ has 3 H. So total H: 5+2+1+1+3=12? Wait, no, maybe I messed up. Wait, the structure is:

Phenyl (C₆H₅) - CH₂ - CH(NHCH₃). So the CH is bonded to CH₂, to N, and to... wait, the CH has one bond to CH₂, one to N, and one to...? Wait, no, the drawing: the CH is a carbon with three bonds: to CH₂, to N, and to...? Wait, no, the structure is:

Benzene ring (C₆H₅) connected to -CH₂- (C₇H₂), then -CH- (C₈H) connected to N (NH) and to a -CH₃ (C₉H₃). So N is bonded to CH (C₈) and to CH₃ (C₉). So:

C: 6 (benzene) + 1 (CH₂) + 1 (CH) + 1 (CH₃) = 9

H: benzene (5) + CH₂ (2) + CH (1) + NH (1) + CH₃ (3) = 5+2+1+1+3=12? Wait, no, NH is N-H, so H from NH is 1, CH₃ is 3 H, CH has 1 H, CH₂ has 2 H, benzene has 5 H. So total H: 5+2+1+1+3=12? Wait, no, maybe the CH is bonded to CH₂, N, and a methyl? Wait, no, the structure is:

The correct molecular formula for methamphetamine is C₁₀H₁₅N? Wait, no, wait, methamphetamine's molecular formula is C₁₀H₁₅N? Wait, no, let's check: methamphetamine is (S)-N-methyl-1-phenylpropan-2-amine, which has formula C₁₀H₁₅N. Wait, so empirical formula is the simplest ratio. C₁₀H₁₅N: divide by GCD(10,15,1)=5? No, GCD(10,15,1) is 1? Wait, 10,15,1: GCD is 1. Wait, no, wait, 10 and 15 have GCD 5. Wait, 10/5=2, 15/5=3, 1/1=1. So empirical formula would be C₂H₃N? No, wait, no, wait, methamphetamine's molecular formula is C₁₀H₁₅N. So to find empirical formula, we divide each subscript by the greatest common divisor (GCD) of 10,15,1. The GCD of 10,15,1 is 1. Wait, no, 10 and 15 have GCD 5, but N is 1, so GCD is 1. Wait, that can't be. Wait, maybe I counted wrong. Wait, let's check the structure again.

Wait, the structure in the image: benzene ring (C₆H₅), then -CH₂-CH(NHCH₃). Wait, the CH is attached to N and to a CH₃, and N is attached to H. So the atoms:

C: 6 (benzene) + 1 (CH₂) + 1 (CH) + 1 (CH₃) + 1 (NHCH₃? No, N is attached to CH and to CH₃. Wait, no, the correct structure of methamphetamine is: C₆H₅ - CH₂ - CH(NHCH₃). Wait, no, it's C₆H₅ - CH₂ - CH(NHCH₃) is wrong. The correct structure is C₆H₅ - CH₂ - CH(CH₃) - NHCH₃. Wait, that's C₁₀H₁₅N. Let's count:

C: 6 (benzene) + 1 (CH₂) + 1 (CH) + 1 (CH₃) + 1 (CH₃) = 6+1+1+1+1=10? Wait, no, CH₂, CH, and two CH₃? Wait, no, methamphetamine is N-methyl-1-phenylpropan-2-amine, which is:

Phenyl (C₆H₅) - CH₂ - CH(CH₃) - NH - CH₃. So that's:

C: 6 (phenyl) + 1 (CH₂) + 1 (CH) + 1 (CH₃) + 1 (CH₃) = 6+1+1+1+1=10

H: 5 (phenyl) + 2 (CH₂) + 1 (CH) + 3 (CH₃) + 1 (NH) + 3 (CH₃) = 5+2+1+3+1+3=15

N: 1

So molecular formula is C₁₀H₁₅N. Now, to find empirical formula, we divide each subscript by the GCD of 10,15,1. The GCD of 10,15,1 is 1, so empirical formula is the same as molecular formula? Wait, no, that can't be. Wait, 10,15,1: GCD is 1, so empirical formula is C₁₀H₁₅N? But that seems odd. Wait, maybe I made a mistake. Wait, no, let's check the molecular formula of methamphetamine: yes, methamphetamine has the molecular formula C₁₀H₁₅N, and since 10,15, and 1 have no common divisor other than 1, the empirical formula is also C₁₀H₁₅N? Wait, no, that's not right. Wait, 10 and 15 are divisible by 5, but N is 1, so we can't divide N by 5. So the GCD is 1. Therefore, the empirical formula is C₁₀H₁₅N.

Wait, but let's confirm. The empirical formula is the simplest whole number ratio of atoms. So if the molecular formula is C₁₀H₁₅N, then since 10,15,1 have GCD 1, the empirical formula is C₁₀H₁₅N.

Step2: Determine empirical formula

Since the molecular formula is C₁₀H₁₅N, and the GCD of 10, 15, and 1 is 1, we divide each subscript by 1, resulting in the same formula.

Answer:

$\boxed{C_{10}H_{15}N}$