Question

(c.6d) relative abundance and atomic mass of the naturally occurring isotopes of hafnium: 5% - 176 amu 19% - 177 amu 27% - 178 amu 14% - 179 amu 35% - 180 amu what is the average atomic mass of hafnium? 177 amu 180 amu 179 amu 176 amu

Explanation:

Step1: Convert percentages to decimals

Convert each percentage to a decimal by dividing by 100.

• \( 5\% = 0.05 \)
• \( 19\% = 0.19 \)
• \( 27\% = 0.27 \)
• \( 14\% = 0.14 \)
• \( 35\% = 0.35 \)

Step2: Multiply each isotope mass by its decimal abundance

• For 176 amu: \( 176 \times 0.05 = 8.8 \)
• For 177 amu: \( 177 \times 0.19 = 33.63 \)
• For 178 amu: \( 178 \times 0.27 = 48.06 \)
• For 179 amu: \( 179 \times 0.14 = 25.06 \)
• For 180 amu: \( 180 \times 0.35 = 63 \)

Step3: Sum the products

Add up all the results from Step 2:
\( 8.8 + 33.63 + 48.06 + 25.06 + 63 = 178.55 \) (approx 179 when rounded, but let's check the options. Wait, maybe miscalculation? Wait, 8.8+33.63=42.43; 42.43+48.06=90.49; 90.49+25.06=115.55; 115.55+63=178.55. Wait, the options have 179. Maybe due to rounding. Wait, let's recalculate:

176*0.05 = 8.8

1770.19: 1770.2=35.4, minus 177*0.01=1.77, so 35.4-1.77=33.63

1780.27: 1780.2=35.6, 178*0.07=12.46, total 35.6+12.46=48.06

1790.14: 1790.1=17.9, 179*0.04=7.16, total 17.9+7.16=25.06

180*0.35=63

Now sum: 8.8 + 33.63 = 42.43; 42.43 + 48.06 = 90.49; 90.49 + 25.06 = 115.55; 115.55 + 63 = 178.55. Wait, the options are 177, 180, 179, 176. 178.55 is closest to 179. So the answer should be 179 amu.

Wait, maybe I made a mistake in calculation? Let's check again:

5% of 176: 0.05*176=8.8

19% of 177: 0.19177. Let's calculate 1770.1=17.7, 177*0.09=15.93, so 17.7+15.93=33.63

27% of 178: 0.27178. 1780.2=35.6, 178*0.07=12.46, 35.6+12.46=48.06

14% of 179: 0.14179. 1790.1=17.9, 179*0.04=7.16, 17.9+7.16=25.06

35% of 180: 0.35*180=63

Now sum: 8.8 + 33.63 = 42.43; 42.43 + 48.06 = 90.49; 90.49 + 25.06 = 115.55; 115.55 + 63 = 178.55. So approximately 179 (since 178.55 is closer to 179 than 178). So the correct option is 179 amu.

Answer:

179 amu (corresponding to the option: 179 amu)