Question

Carry out the following calculations, express the answers to the correct numbers of significant figures, and include units in the answers. (a) \(5280 \frac{\mathrm{m}}{\mathrm{km}} \times 6.2 \mathrm{~km}\) (b) \(4.5 \mathrm{~m} \times 3.25 \mathrm{~m}\) (c) \(2.50 \mathrm{~g} \div 8.3 \frac{\mathrm{g}}{\mathrm{cm}^{3}}\) (d) \(4.70 \mathrm{~cm} \times 6.8 \mathrm{~cm} \times 2.54 \mathrm{~cm}\)

Short answer

(a) \(3.3 \times 10^4 \mathrm{~m}\); (b) \(15 \mathrm{~m}^2\); (c) \(0.30 \mathrm{~cm}^3\); (d) \(81 \mathrm{~cm}^3\).

Step-by-step solution

Understand the Significant Figures

Review the concepts of significant figures, understanding that the number of significant figures in a product or quotient should be the same as the number in the measurement with the fewest significant figures.

Solve Part (a)

Calculate the multiplication: \(5280 \frac{\mathrm{m}}{\mathrm{km}} \times 6.2 \mathrm{~km} = 32736 \mathrm{~m}\). The result should be expressed to two significant figures because 6.2 km has two significant figures. Therefore, the answer is \(3.3 \times 10^4 \mathrm{~m}\).

Solve Part (b)

Calculate the multiplication: \(4.5 \mathrm{~m} \times 3.25 \mathrm{~m} = 14.625 \mathrm{~m}^2\). The result should be expressed to two significant figures since 4.5 m has two significant figures. Therefore, the answer is \(15 \mathrm{~m}^2\).

Solve Part (c)

Calculate the division: \(2.50 \mathrm{~g} \div 8.3 \frac{\mathrm{g}}{\mathrm{cm}^{3}} = 0.3012048193 \mathrm{~cm}^3\). The result should be expressed to two significant figures as 8.3 \(\mathrm{g/cm}^3\) has the fewest significant figures with two. Therefore, the answer is \(0.30 \mathrm{~cm}^3\).

Solve Part (d)

Calculate the multiplication: \(4.70 \mathrm{~cm} \times 6.8 \mathrm{~cm} \times 2.54 \mathrm{~cm} = 81.1224 \mathrm{~cm}^3\). The result should be expressed to two significant figures since 6.8 cm has the fewest significant figures. Therefore, the answer is \(81 \mathrm{~cm}^3\).

Key concepts

Multiplication in Chemistry

Multiplication in chemistry involves dealing with measurements that carry units, such as meters, kilometers, or grams. When you multiply two measurements, the units are also multiplied, adding another layer of complexity to calculations. Here's how it works:

* Start by lining up your numeric values. Multiply them as if they were simple numbers. For example, if you multiply 6.2 km by 5280 m/km, first focus solely on the 6.2 and 5280, the numbers. Ignore the units initially. * Next, take into account the units. By definition, a kilometer contains 1000 meters (unit conversion in action). So when you multiply km by m/km, the km units cancel out, leaving you with meters. Always remember to consider significant figures. Significant figures are the digits in a measurement that contribute to its precision. In multiplication, your final answer's number of significant figures should match the number with the least significant figures from the values you multiplied. This ensures your answer is both precise and reliable.

For instance, in multiplying 4.5 m by 3.25 m, even though the result 14.625 seems longer, the final answer should be trimmed to two significant figures, resulting in 15 m². This preserves the precision, as 4.5 has two significant figures, dictating the precision of the result.

Division in Chemistry

Division in chemistry works similarly to multiplication, except instead of multiplying, you are dividing measurements to find out how many times one value fits into another. Let's break it down:

* Begin by dividing the numbers, ignoring the units for the moment. For example, dividing 2.50 grams by 8.3 grams per cubic centimeter results in approximately 0.3012. * Now consider the units. Here, you are dividing grams by grams/cm³, which simplifies to cubic centimeters. This is because dividing by a fraction (g/cm³) is the same as multiplying by its reciprocal (cm³/g), resulting in cubic centimeters. When dealing with significant figures in division, apply the same rule as with multiplication: the result should have the same number of significant figures as the measurement with the fewest significant figures used in the calculation.

In the example of 2.50 g being divided by 8.3 g/cm³, the 8.3 has two significant figures, which makes your result 0.30 cm³. This rule maintains consistency and precision throughout division operations in chemistry.

Unit Conversion in Measurements

Unit conversion is a process often needed in chemistry, especially when combining or transforming measurements that are initially in different units. The key to successful unit conversion is understanding conversion factors.

* A conversion factor is a ratio that expresses how many of one unit relates to another, like how 1 kilometer equals 1000 meters. * Set up your conversion factor such that the units you want to eliminate are canceled out. For example, to convert kilometers to meters in the calculation 5280 m/km multiplied by 6.2 km, set up 5280 m/km correctly to ensure the km units are canceled, leaving you with meters. When executing unit conversions, ensure that your final answer reflects correct significant figures and unit clarity. Units work almost like algebraic variables that get canceled or combined during arithmetic operations.

Proper unit conversion is crucial to get accurate results in chemistry, ensuring measurements are in a standard format for comparison or further calculations. Remember: keep track of both significant figures and units at each step to sustain integrity in your scientific work.