Question

There are 26,900,000,000,000,000,000,000 atoms in 1 liter of argon gas at standard temperature and pressure. Express this number in scientific notation.

Short answer

The number is expressed as \( 2.69 \times 10^{22} \) in scientific notation.

Step-by-step solution

Understand Scientific Notation

Scientific notation is a way to express very large or very small numbers by using powers of ten. The format is \( a \times 10^n \), where \( 1 \leq a < 10 \) and \( n \) is an integer.

Identify Coefficient and Exponent

To convert 26,900,000,000,000,000,000,000 atoms into scientific notation, we need to express this number as \( a \times 10^n \). Start by identifying \( a \), which should be the original number with a single digit before the decimal point.

Convert to Decimal Form

Write the number by moving the decimal point such that the coefficient \( a \) becomes 2.69. Original number: 26,900,000,000,000,000,000,000 becomes 2.69 when we place the decimal after the first non-zero digit (2).

Determine the Power of Ten

Count how many places the decimal has moved from its original position to its final position in 2.69. It has moved 22 places to the left. Therefore, the exponent \( n \) is 22.

Express in Scientific Notation

Combine the coefficient and exponent to express the number in scientific notation: \( 2.69 \times 10^{22} \).

Key concepts

Powers of Ten

Scientific notation is a method used to express very large or very small numbers conveniently using powers of ten. This format uses a base multipled by a power of ten. For example, you regularly encounter numbers like 1,000 or 0.0001. When expressed in scientific notation, these numbers become easier to handle, such as:

• 1,000 is written as \(1 \times 10^3\)
• 0.0001 becomes \(1 \times 10^{-4}\)

In scientific notation, each number is written as \( a \times 10^n \). Here, \(a\) is called the coefficient and must be a number between 1 and 10, and \(n\) is the exponent, showing the number of times you multiply \(a\) by ten.

To convert a large number like 26,900,000,000,000,000,000,000 into scientific notation, it becomes important to properly select \(a\) and \(n\). Move the decimal point in the number until you have a coefficient \(a\) between 1 and 10. Count each move as a step in units of ten, and that becomes your exponent \(n\).

By following this process, you ensure the number is succinctly represented, making it easier to manage mathematically.

Standard Temperature and Pressure

In many scientific studies, especially in chemistry and physics, you will often encounter the term "Standard Temperature and Pressure" (STP). This is a predefined set of conditions used to enable consistent experimental results and data comparisons.

At STP, the temperature is set at 273.15 K (0°C or 32°F), and the pressure is set at 1 atmosphere (101.3 kPa). Under these standard conditions, gases behave more predictably, which aids scientists when calculating and comparing the properties of gases.

STP is useful because:

• It allows scientists to compare different gases under uniform conditions
• It provides a baseline for calculating volumes, densities, and molecular speeds of gases
• It simplifies the mathematical equations used in gas laws, like the Ideal Gas Law, for computational ease

So, when asked about quantities such as the number of atoms in a gas at STP, you can be certain of the exact conditions you're referring to, hence making your calculations accurate and reproducible.

Atoms in a Gas

Understanding the concept of atoms in a gas requires a basic appreciation of how gases behave under different conditions. At the molecular level, gases consist of a large number of tiny particles, typically atoms or molecules, that are spaced far apart compared to their size. This allows them to freely move and fill the volume of their container uniformly.

When dealing with gases, the number of atoms and their interactions are calculated using Avogadro's number, which is approximately \(6.022 \times 10^{23}\) atoms (or molecules) per mole. This helps convert the number of moles of a gas to a palpable number of atoms.

In real-world applications, it's not feasible to count each atom of a gas one by one, especially at standard temperature and pressure, where gas volumes can be large. Therefore, using scientific notation and understanding gas laws enables us to handle these huge numbers easily. For instance, counting the number of atoms in a liter of gas—like argon at STP—involves using systematic approximation and clear labeling of decimal movement in calculations.

Grasping this concept aids in comprehensive learning and practical application, as gases' behaviors are crucial in various fields such as meteorology, engineering, and environmental science.