Mass Of 1000 Gold Atoms In Scientific Notation Calculation

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In the fascinating world of chemistry, understanding the incredibly small masses of atoms is crucial. When dealing with such minuscule values, scientific notation becomes an indispensable tool. This article delves into a fundamental problem involving the mass of gold atoms, providing a step-by-step explanation and highlighting the importance of scientific notation in simplifying complex calculations. By exploring this specific example, we aim to illuminate the broader applications of scientific notation in various scientific disciplines.

The problem at hand is a classic example of how scientific notation helps manage very small numbers. We are given that a single atom of gold has a mass of approximately 3.25 × 10⁻²² grams. The challenge is to determine the mass of 1000 such atoms and express the result in scientific notation. This task not only requires basic multiplication skills but also a clear understanding of how exponents work when dealing with scientific notation.

To find the mass of 1000 gold atoms, we need to multiply the mass of a single gold atom by 1000. Mathematically, this can be represented as:

Mass of 1000 gold atoms = (Mass of 1 gold atom) × 1000

Substituting the given mass of a single gold atom, we get:

Mass of 1000 gold atoms = (3.25 × 10⁻²² grams) × 1000

Here, 1000 can be expressed as 10³, which simplifies our calculation:

Mass of 1000 gold atoms = (3.25 × 10⁻²²) × 10³ grams

Now, we apply the rules of exponents, which state that when multiplying numbers with the same base, we add the exponents:

Mass of 1000 gold atoms = 3.25 × 10⁽⁻²²⁺³⁾ grams

Let's walk through the calculation step by step to ensure clarity:

  1. Identify the given values:
    • Mass of 1 gold atom = 3.25 × 10⁻²² grams
    • Number of atoms = 1000
  2. Express 1000 in scientific notation:
    • 1000 = 10³
  3. Multiply the mass of one gold atom by 1000:
    • (3.25 × 10⁻²²) × 10³
  4. Apply the exponent rule (aᵐ × aⁿ = aᵐ⁺ⁿ):
      1. 25 × 10⁽⁻²²⁺³⁾
  5. Add the exponents:
      1. 25 × 10⁻¹⁹ grams

Therefore, the mass of 1000 gold atoms is 3.25 × 10⁻¹⁹ grams.

Scientific notation is a way of expressing numbers that are either very large or very small in a compact and easily manageable form. It is particularly useful in scientific disciplines where dealing with such numbers is common. A number in scientific notation is expressed as:

N × 10ⁿ

Where:

  • N is a number between 1 and 10 (but less than 10)
  • 10 is the base
  • n is an integer exponent

For instance, the mass of a single gold atom, 3.25 × 10⁻²² grams, is written in scientific notation. Here, 3.25 is the number between 1 and 10, and -22 is the exponent. The negative exponent indicates that the number is very small, specifically, 3.25 divided by 10 to the power of 22.

Benefits of Using Scientific Notation

Using scientific notation offers several advantages, especially in scientific and mathematical contexts:

  • Simplifies Complex Calculations: Scientific notation makes it easier to perform arithmetic operations on very large or very small numbers. For instance, multiplying or dividing numbers in scientific notation involves simple operations on the exponents.
  • Reduces Errors: Writing out very large or very small numbers in their full decimal form can be cumbersome and prone to errors. Scientific notation reduces the chances of making mistakes by providing a more concise representation.
  • Enhances Clarity: Scientific notation clearly indicates the magnitude of a number. The exponent provides a quick indication of whether the number is very large or very small.
  • Facilitates Comparisons: It is easier to compare numbers when they are expressed in scientific notation. By comparing the exponents, one can quickly determine the relative sizes of the numbers.

Scientific notation is not just a theoretical concept; it has numerous practical applications in various fields:

  • Chemistry: As demonstrated in the problem, chemistry often deals with extremely small quantities, such as the masses of atoms and molecules. Scientific notation is essential for expressing these values.
  • Physics: Physics involves a wide range of scales, from the subatomic to the cosmic. Scientific notation is used to express quantities such as the speed of light (approximately 3 × 10⁸ meters per second) and the gravitational constant (approximately 6.674 × 10⁻¹¹ N(m/kg)²).
  • Astronomy: Astronomers deal with vast distances and masses. The distance to the nearest star, Proxima Centauri, is about 4.02 × 10¹⁶ meters. The mass of the Sun is approximately 1.989 × 10³⁰ kilograms. Scientific notation is indispensable in this field.
  • Computer Science: Computer scientists use scientific notation to describe storage capacities and processing speeds. For example, a computer's storage capacity might be expressed in terabytes (TB), where 1 TB is approximately 1 × 10¹² bytes.
  • Engineering: Engineers use scientific notation in various calculations, such as determining structural loads, electrical currents, and material properties.

When working with scientific notation, it is important to avoid common mistakes to ensure accurate results:

  • Incorrectly Placing the Decimal Point: The number N in scientific notation should always be between 1 and 10 (but less than 10). For example, 32.5 × 10⁻²³ is incorrect; it should be written as 3.25 × 10⁻²².
  • Miscalculating the Exponent: When multiplying or dividing numbers in scientific notation, ensure that the exponents are added or subtracted correctly. A mistake in the exponent can lead to a significant error in the result.
  • Forgetting the Units: Always include the appropriate units with the final answer. For example, in our problem, the answer is 3.25 × 10⁻¹⁹ grams, not just 3.25 × 10⁻¹⁹.
  • Not Adjusting the Exponent After Multiplication or Division: After performing multiplication or division, the resulting number might not be in the correct scientific notation form. Adjust the decimal point and the exponent accordingly.

To reinforce your understanding of scientific notation, try these practice problems:

  1. The mass of a hydrogen atom is approximately 1.67 × 10⁻²⁴ grams. What is the mass of 1 million hydrogen atoms?
  2. The speed of light is approximately 3 × 10⁸ meters per second. How far does light travel in 1 hour?
  3. A certain bacterium has a length of 2.5 × 10⁻⁶ meters. If 10,000 bacteria are laid end to end, what is the total length?

In summary, determining the mass of 1000 gold atoms from the mass of a single atom is a practical application of scientific notation. By multiplying 3.25 × 10⁻²² grams by 1000 (or 10³), we found the mass to be 3.25 × 10⁻¹⁹ grams. Scientific notation simplifies the handling of very small or very large numbers, making it an invaluable tool in chemistry, physics, and other scientific disciplines. Understanding and using scientific notation correctly is essential for accurate and efficient scientific calculations.

This problem not only demonstrates a basic calculation but also highlights the broader significance of scientific notation in managing and interpreting quantitative data. By mastering these fundamental concepts, students and professionals alike can confidently tackle more complex scientific challenges.

The mass of 1000 gold atoms is 3.25 × 10⁻¹⁹ grams.