Problem 55 - Chemistry: An Atoms-Focused Approach

Question

At a distance of 4.3 light-years, Proxima Centauri is the nearest star to our solar system. What is the distance to Proxima Centauri in kilometers? (The speed of light in space is \(2.998 \times 10^{8} \mathrm{m} / \mathrm{s} .\) )

Step-by-Step Solution

Answer

Answer: To determine the distance to Proxima Centauri in kilometers, use the following expression: $$ \text{Distance (km)} = \frac{4.3 \, \text{light-years} \times 2.998 \times 10^{8} \mathrm{m/s} \times 365.25 \times 24 \times 60 \times 60 \, \text{seconds}}{1000} $$ Calculating this value gives the distance to Proxima Centauri in kilometers.

Step 1: Convert light-years to meters

We are given the distance to Proxima Centauri as 4.3 light-years. As the speed of light is \(2.998 \times 10^{8} \mathrm{m} / \mathrm{s}\), we can find the distance traveled by light in one year as follows: $$ 1 \text{ light-year} = 2.998 \times 10^{8} \mathrm{m/s} \times \text{Number of seconds in a year} $$ Using the fact there are 365.25 days in a year, and 24 hours in a day, 60 minutes in an hour, and 60 seconds in a minute, we find: $$ 1 \text{ light-year} = 2.998 \times 10^{8} \mathrm{m/s} \times 365.25 \times 24 \times 60 \times 60 \, \text{seconds} $$

Step 2: Calculate distance in meters

Now, we can find the distance to Proxima Centauri in meters by multiplying the given distance in light-years with the value of one light-year in meters: $$ \text{Distance (m)} = 4.3 \, \text{light-years} \times 2.998 \times 10^{8} \mathrm{m/s} \times 365.25 \times 24 \times 60 \times 60 \, \text{seconds} $$

Step 3: Convert meters to kilometers

To convert the calculated distance from meters into kilometers, we simply divide by the conversion factor of 1000: $$ \text{Distance (km)} = \frac{\text{Distance (m)}}{1000} $$ Substituting the expression from Step 2: $$ \text{Distance (km)} = \frac{4.3 \, \text{light-years} \times 2.998 \times 10^{8} \mathrm{m/s} \times 365.25 \times 24 \times 60 \times 60 \, \text{seconds}}{1000} $$ Now you can calculate the distance to Proxima Centauri in kilometers using the above expression.

Key Concepts

Light-Year Conversion

A light-year is a common unit for measuring astronomical distances. It's defined as the distance that light travels in one year. Since light moves remarkably fast, this distance is vast. To convert light-years into a more relatable metric such as meters, we must first understand how much light travels within that time span.
Light covers approximately 299,792,458 meters per second. When multiplied by the number of seconds in a year, you get the total distance light travels in a year. Here's the process:

• The number of seconds in a year = 365.25 days/year × 24 hours/day × 60 minutes/hour × 60 seconds/minute.
• Calcuate: 1 light-year = Speed of light \(2.998 \times 10^{8} \text{ m/s} \times \text{Total seconds in a year}\)

This calculation will give the distance in meters for one light-year. Knowing this, you can easily multiply by the number of light-years to measure astronomical distances like the distance to stars.

Speed of Light

The speed of light is a fundamental constant used in many scientific fields, especially in physics and astronomy. It's represented as \(c\) with a value of \(2.998 \times 10^{8}\) meters per second. This constant plays a critical role in understanding how light travels through the vacuum of space.
When converting the distance of celestial objects from light-years to another unit like kilometers, astronomers rely on this constant speed. It's instrumental in calculating how far light has traveled over a specified period, such as one year (hence the term light-year).
Important points to remember about the speed of light:

• It's the maximum speed at which all energy, particles, and information in the universe can travel.
• The constancy of light's speed also underlies Albert Einstein's theory of relativity.

Understanding the speed of light allows us to comprehend not just spatial distances but also the time it takes for information (like light from a distant star) to reach us on Earth.

Unit Conversion

Unit conversion is a crucial mathematical process in science, allowing us to make sense of different scales of measurement. This is essential when working with astronomical distances, particularly in translating light-years into kilometers, which are more commonly used on Earth.
After calculating the distance in meters, which was determined using the speed of light, the next step is converting meters into kilometers. This is straightforward. Since 1 kilometer equals 1000 meters, you divide the distance in meters by 1000 to get the distance in kilometers.
The steps for converting meters into kilometers are:

• Compute the total distance in meters: \( \text{Distance (m)} = \text{Light-year conversion} \times \text{Number of light-years} \)
• Apply the conversion: \( \text{Distance (km)} = \frac{\text{Distance (m)}}{1000} \)

Consequently, these conversions make astronomical distances more comprehensible and manageable when considering terrestrial scales, helping to bridge the gap between cosmic measurements and everyday human understanding.