Problem 14 - Chemistry Matter and Change

Question

Challenge Express each answer in scientific notation in the units indicated. a. \(\left(1.26 \times 10^{4} \mathrm{kg}\right)+\left(2.5 \times 10^{6} \mathrm{g}\right)\) in kg b. \((7.06 \mathrm{g})+\left(1.2 \times 10^{-4} \mathrm{kg}\right)\) in \(\mathrm{kg}\) c. \(\left(4.39 \times 10^{5} \mathrm{kg}\right)-\left(2.8 \times 10^{7} \mathrm{g}\right)\) in \(\mathrm{kg}\) d. \(\left(5.36 \times 10^{-1} \mathrm{kg}\right)-\left(7.40 \times 10^{-2} \mathrm{kg}\right)\) in \(\mathrm{g}\)

Step-by-Step Solution

Answer

a. \(1.51 \times 10^{4} \text{ kg}\); b. \(7.18 \times 10^{-3} \text{ kg}\); c. \(4.11 \times 10^{5} \text{ kg}\); d. \(4.62 \times 10^{2} \text{ g}\).

Step 1a: Convert Units

First, we need to convert the grams to kilograms. Since there are 1000 grams in a kilogram, we convert the grams to kilograms: \[2.5 \times 10^{6} \text{ g} = \frac{2.5 \times 10^{6}}{1000} \text{ kg} = 2.5 \times 10^{3} \text{ kg}\]

Step 2a: Add the Masses

Now add the two masses, both in kilograms:\[(1.26 \times 10^{4} \text{ kg}) + (2.5 \times 10^{3} \text{ kg}) = 1.26 \times 10^{4} \text{ kg} + 2.5 \times 10^{3} \text{ kg} = 1.51 \times 10^{4} \text{ kg}\]

Step 1b: Convert Units

Convert the grams to kilograms: \[7.06 \text{ g} = \frac{7.06}{1000} \text{ kg} = 7.06 \times 10^{-3} \text{ kg}\]

Step 2b: Add the Masses

Now add the two masses, both in kilograms:\[(7.06 \times 10^{-3} \text{ kg}) + (1.2 \times 10^{-4} \text{ kg}) = 7.06 \times 10^{-3} \text{ kg} + 1.2 \times 10^{-4} \text{ kg} = 7.18 \times 10^{-3} \text{ kg}\]

Step 1c: Convert Units

Convert the grams to kilograms: \[2.8 \times 10^{7} \text{ g} = \frac{2.8 \times 10^{7}}{1000} \text{ kg} = 2.8 \times 10^{4} \text{ kg}\]

Step 2c: Subtract the Masses

Subtract the two masses, both in kilograms:\[(4.39 \times 10^{5} \text{ kg}) - (2.8 \times 10^{4} \text{ kg}) = 4.39 \times 10^{5} \text{ kg} - 2.8 \times 10^{4} \text{ kg} = 4.11 \times 10^{5} \text{ kg}\]

Step 1d: Subtract the Masses

This time, both masses are already in kilograms, so directly subtract the two values:\[(5.36 \times 10^{-1} \text{ kg}) - (7.40 \times 10^{-2} \text{ kg}) = 5.36 \times 10^{-1} \text{ kg} - 7.40 \times 10^{-2} \text{ kg} = 4.62 \times 10^{-1} \text{ kg}\]

Step 2d: Convert Final Answer to Grams

Convert the resulting kilogram value to grams by multiplying by 1000:\[4.62 \times 10^{-1} \text{ kg} \times 1000 = 4.62 \times 10^{2} \text{ g}\]

Key Concepts

Unit Conversion

Understanding unit conversion is essential in scientific calculations, especially when you're dealing with different metrics. A common example in science is converting between grams and kilograms. This conversion is straightforward because both units are part of the metric system, with one kilogram equaling 1,000 grams. Here's how you do it:

• To convert grams to kilograms, divide the number of grams by 1,000.
• To convert kilograms to grams, multiply the number of kilograms by 1,000.

This is vital when adding or subtracting masses that are not initially in the same unit, allowing you to perform calculations with uniform units. For example, converting grams into kilograms before performing operations helps maintain consistency and accuracy in your results.

Mass Addition

To add masses together in scientific calculations, make sure they are in the same unit first. This prevents mismatch and errors in your results. Once your units match, simply add the values as you would in any addition operation. For instance, if you have masses of \(1.26 \times 10^{4} \text{ kg}\) and \(2.5 \times 10^{3} \text{ kg}\), ensure they are both in kilograms before addition. The equation becomes:

• \((1.26 \times 10^{4} \text{ kg}) + (2.5 \times 10^{3} \text{ kg})\)

This results in \(1.51 \times 10^{4} \text{ kg}\), a simple exercise in matching units and performing addition in scientific notation. Remember, scientific notation is helpful for handling very large or small numbers, keeping your calculations clear and manageable.

Mass Subtraction

Subtracting masses works similarly to addition, where consistency in unit is key. Convert all masses into the same unit beforehand and apply proper mathematical operations.Consider an example where you have \(4.39 \times 10^{5} \text{ kg}\) and you want to subtract \(2.8 \times 10^{7} \text{ g}\). To proceed, convert \(2.8 \times 10^{7} \text{ g}\) to kilograms, resulting in \(2.8 \times 10^{4} \text{ kg}\). Now, both masses are in kilograms:

• \((4.39 \times 10^{5} \text{ kg}) - (2.8 \times 10^{4} \text{ kg})\)

The subtraction yields \(4.11 \times 10^{5} \text{ kg}\). This operation demonstrates subtracting using scientific notation, maintaining accuracy without being overwhelmed by large number manipulations.

Metric System

The metric system is widely used in science and mathematics due to its simplicity and universality. Its base-10 structure makes conversions between different units convenient and intuitive. For example, changing between grams and kilograms is a matter of multiplying or dividing by 1,000. Within the metric system, units of mass include:

• Grams (g), which is the base unit for mass.
• Kilograms (kg), where 1 kilogram equals 1,000 grams.

Adopting the metric system for scientific calculations allows for straightforward conversions and clear communication of results globally. Understanding this system can significantly ease the process of working with scientific notation and various unit conversions, making it an indispensable tool in the world of science and mathematics.