AP Chemistry practice worksheet focusing on significant figures, covering identification, rounding, and calculation rules.

Problem Analysis:

The document contains three sections related to significant figures (sig figs), which are crucial in scientific measurements. Let's solve each section step by step.

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Section 1: State the number of significant digits in each measurement

#### Rules for Significant Figures:
1. Non-zero digits are always significant.
2. Zeros between non-zero digits are significant.
3. Leading zeros are not significant.
4. Trailing zeros are significant if they come after a decimal point or are followed by a decimal point.
5. In scientific notation, only the digits in the coefficient count as significant.

#### Solutions:
- a. 2804 m: All digits are significant.
Answer: 4 sig figs

- b. 2.84 km: All digits are significant.
Answer: 3 sig figs

- c. 5.029 m: All digits are significant.
Answer: 4 sig figs

- d. 0.003068 m: Leading zeros are not significant; trailing zeros after the decimal are significant.
Answer: 4 sig figs

- e. 4.6 × 10³ m: Only the digits in the coefficient (4.6) are significant.
Answer: 2 sig figs

- f. 4.060 × 10⁻⁵ m: All digits in the coefficient are significant.
Answer: 4 sig figs

- g. 750 m: The trailing zero is not significant because it is not after a decimal point.
Answer: 2 sig figs

- h. 750. m: The trailing zero is significant because it is after a decimal point.
Answer: 3 sig figs

- i. 75,000 m: The trailing zeros are not significant because they are not after a decimal point.
Answer: 2 sig figs

- j. 75.00 m: All digits are significant, including the trailing zeros after the decimal point.
Answer: 4 sig figs

- k. 75,000.0 m: The trailing zeros after the decimal point are significant.
Answer: 6 sig figs

- l. 10 cm: The trailing zero is not significant because it is not after a decimal point.
Answer: 1 sig fig

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Section 2: Round the following numbers as indicated

#### Rules for Rounding:
1. Identify the digit to be rounded based on the instruction (e.g., to four sig figs, to the tenths place, etc.).
2. Look at the digit immediately to the right of the rounding digit:
- If it is 5 or greater, round up.
- If it is less than 5, keep the rounding digit the same.

#### Solutions:
- To four sig figs:
- 3.682417: The fourth sig fig is 2. The next digit is 4 (less than 5), so no rounding up.
Answer: 3.682
- 21.860051: The fourth sig fig is 6. The next digit is 0 (less than 5), so no rounding up.
Answer: 21.86
- 112.511: The fourth sig fig is 5. The next digit is 1 (less than 5), so no rounding up.
Answer: 112.5
- 45.4673: The fourth sig fig is 6. The next digit is 7 (greater than 5), so round up.
Answer: 45.47

- To the tenths place:
- 2.473: The tenths place is 4. The next digit is 7 (greater than 5), so round up.
Answer: 2.5
- 5.687524: The tenths place is 6. The next digit is 8 (greater than 5), so round up.
Answer: 5.7
- 7.555: The tenths place is 5. The next digit is 5 (equal to 5), so round up.
Answer: 7.6
- 8.235: The tenths place is 2. The next digit is 3 (less than 5), so no rounding up.
Answer: 8.2

- To the hundredths place:
- 79.2588: The hundredths place is 5. The next digit is 8 (greater than 5), so round up.
Answer: 79.26
- 0.03062: The hundredths place is 0. The next digit is 0 (less than 5), so no rounding up.
Answer: 0.03
- 3.4125: The hundredths place is 1. The next digit is 2 (less than 5), so no rounding up.
Answer: 3.41
- 41.86632: The hundredths place is 6. The next digit is 6 (greater than 5), so round up.
Answer: 41.87

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Section 3: Solve the following problems and report answers with appropriate number of significant digits

#### Rules for Arithmetic Operations:
1. Addition/Subtraction: The result should have the same number of decimal places as the least precise measurement.
2. Multiplication/Division: The result should have the same number of significant figures as the least precise measurement.

#### Solutions:
- a. 6.201 cm + 7.4 cm + 0.68 cm + 12.0 cm =
Least precise measurement: 7.4 cm (1 decimal place).
Sum: \( 6.201 + 7.4 + 0.68 + 12.0 = 26.281 \).
Round to 1 decimal place: 26.3 cm.

- b. 1.6 km + 1.62 m + 1200 cm =
Convert all units to meters:
\( 1.6 \text{ km} = 1600 \text{ m} \), \( 1.62 \text{ m} = 1.62 \text{ m} \), \( 1200 \text{ cm} = 12 \text{ m} \).
Sum: \( 1600 + 1.62 + 12 = 1613.62 \text{ m} \).
Least precise measurement: 1.6 km (2 sig figs).
Round to 2 sig figs: 1.6 × 10³ m.

- c. 8.264 g - 7.8 g =
Least precise measurement: 7.8 g (1 decimal place).
Difference: \( 8.264 - 7.8 = 0.464 \).
Round to 1 decimal place: 0.5 g.

- d. 10.4168 m - 6.0 m =
Least precise measurement: 6.0 m (2 sig figs).
Difference: \( 10.4168 - 6.0 = 4.4168 \).
Round to 2 sig figs: 4.4 m.

- e. 12.00 m + 15.0018 kg =
Note: Units are different (meters and kilograms), so this operation is invalid.
Answer: Not possible.

- f. 1.319 cm × 2.3 cm =
Least precise measurement: 2.3 cm (2 sig figs).
Product: \( 1.319 \times 2.3 = 3.0337 \).
Round to 2 sig figs: 3.0 cm².

- g. 5.7621 m × 6.201 m =
Least precise measurement: 6.201 m (4 sig figs).
Product: \( 5.7621 \times 6.201 = 35.7167821 \).
Round to 4 sig figs: 35.72 m².

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Final Answers:

1. Number of significant digits:
- a. 4
- b. 3
- c. 4
- d. 4
- e. 2
- f. 4
- g. 2
- h. 3
- i. 2
- j. 4
- k. 6
- l. 1

2. Rounded numbers:
- To four sig figs: 3.682, 21.86, 112.5, 45.47
- To the tenths place: 2.5, 5.7, 7.6, 8.2
- To the hundredths place: 79.26, 0.03, 3.41, 41.87

3. Calculations:
- a. 26.3 cm
- b. 1.6 × 10³ m
- c. 0.5 g
- d. 4.4 m
- e. Not possible
- f. 3.0 cm²
- g. 35.72 m²

Final Boxed Answer:
\[
\boxed{
\begin{array}{l}
\text{1. a. 4, b. 3, c. 4, d. 4, e. 2, f. 4, g. 2, h. 3, i. 2, j. 4, k. 6, l. 1} \\
\text{2. a. 3.682, b. 21.86, c. 112.5, d. 45.47, e. 2.5, f. 5.7, g. 7.6, h. 8.2, i. 79.26, j. 0.03, k. 3.41, l. 41.87} \\
\text{3. a. 26.3 cm, b. 1.6 × 10³ m, c. 0.5 g, d. 4.4 m, e. Not possible, f. 3.0 cm², g. 35.72 m²}
\end{array}
}
\]