Calculating With Significant Figures A Chemistry Problem Solution

by James Vasile 66 views

Hey guys! Ever find yourself staring at a chemistry problem that looks like a jumbled mess of numbers and wonder where to even start? I get it! Significant figures can be tricky, but they're super important in chemistry because they tell us about the precision of our measurements. Let's break down a classic significant figures calculation step by step, so you'll be a pro in no time.

The Problem: 78.01 x 124.8760 / (52.60 - 3.39)

Okay, so here's the problem we're tackling: 78.01 x 124.8760 / (52.60 - 3.39). At first glance, it might seem intimidating, but don't worry, we'll take it one piece at a time. The key to getting the right answer with the correct significant figures is to follow the order of operations (PEMDAS/BODMAS) and apply the significant figure rules at each step. Remember, significant figures indicate the reliable digits in a measurement or calculation, and they're crucial for representing the accuracy of your results.

Step 1: Subtraction (Parentheses First!)

The first thing we need to do is tackle the subtraction within the parentheses: 52.60 - 3.39. When we subtract (or add), we're looking at the number of decimal places. Here's the calculation:

  1. 60
    1. 39

= 49.21

Both 52.60 and 3.39 have two decimal places. So, our result, 49.21, also has two decimal places. In this case, all digits are significant, giving us four significant figures. The rule for addition and subtraction is that the result should have the same number of decimal places as the number with the fewest decimal places.

So, after the subtraction, we now have:

78.01 x 124.8760 / 49.21

Step 2: Multiplication and Division (Left to Right)

Now we move on to multiplication and division. Remember, we perform these operations from left to right. First up, multiplication:

78.01 x 124.8760

When multiplying and dividing, we focus on the total number of significant figures in each number, not the decimal places. 78.01 has four significant figures, and 124.8760 has seven significant figures. When you multiply these two numbers together on a calculator, you get a long number:

9741.53676

But we can't just write down all those digits! We need to round our answer to the same number of significant figures as the number with the fewest significant figures in the calculation. In this case, 78.01 has the fewest (four significant figures), so our intermediate result needs to have four significant figures as well. That means we look at the first four digits (9741) and then consider the next digit (5) to determine how to round. Since it's a 5, we round up:

9742

So, after the multiplication (and rounding to the correct number of significant figures), we have:

9742 / 49.21

Now, let's do the division. We're dividing 9742 (four significant figures) by 49.21 (also four significant figures). When you divide these numbers, you get:

  1. 959347...

Again, we can't just write down all those digits. We need to round to the correct number of significant figures. Since both numbers in our division have four significant figures, our final answer should also have four significant figures. Looking at the first four digits (197.9) and the next digit (5), we round up:

198.0

Significant Figures and Scientific Notation

Before we wrap up, let's touch on scientific notation, which is a handy way to express very large or very small numbers and keep track of significant figures. For example, if we had a result of 200 with only one significant figure, we'd write it as 2 x 10². If we meant 200 with three significant figures, we'd write it as 2.00 x 10². Scientific notation makes it crystal clear how many digits are significant.

Why Significant Figures Matter

Now, you might be wondering, “Why all this fuss about significant figures?” Well, in chemistry (and science in general), it's crucial to represent the precision of your measurements and calculations accurately. Using the correct number of significant figures shows that you understand the limitations of your equipment and methods. It also prevents you from overstating the accuracy of your results.

Imagine you're measuring the volume of a liquid with a graduated cylinder that has markings every 1 mL. You can estimate to the nearest 0.1 mL, but you can't know the volume to the nearest 0.01 mL. Reporting your result with too many digits would give a false impression of accuracy. Significant figures ensure that your results reflect the true uncertainty of your measurements.

The Answer and Why It's Correct

So, after carefully working through the problem and applying the rules of significant figures, we arrive at our final answer: 198.0

This corresponds to answer choice C. This is the correct answer because it has four significant figures, which is consistent with the least number of significant figures in the original calculation (four in 78.01 and 49.21).

Common Mistakes to Avoid

To make sure you nail these problems every time, let's quickly go over some common pitfalls:

  • Forgetting the Order of Operations: Always follow PEMDAS/BODMAS (Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction).
  • Rounding Too Early: Try to keep extra digits during intermediate calculations and only round at the very end to avoid accumulating rounding errors.
  • Miscounting Significant Figures: Remember the rules for zeros (leading zeros are never significant, trailing zeros are significant only if there's a decimal point, etc.).
  • Ignoring Significant Figures in Addition/Subtraction: Don't forget that for addition and subtraction, you focus on decimal places, not the total number of significant figures.

Practice Makes Perfect

The best way to master significant figures is to practice! Work through plenty of examples, and don't be afraid to make mistakes – that's how we learn. Pay close attention to the rules, and you'll become a significant figure whiz in no time.

I hope this breakdown helped you understand how to tackle significant figure calculations. Keep practicing, and you'll be a chemistry pro before you know it! If you have any more questions, feel free to ask. Happy calculating!