5.12 X 6.802: Significant Figures Explained

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Let's dive into a common type of chemistry question: multiplication involving significant figures. Guys, this stuff is super important in chemistry because it ensures that our calculations reflect the precision of our measurements. No fudging the data, alright? When you're multiplying numbers, the result should have the same number of significant figures as the number with the fewest significant figures. Keep that in mind, it's a golden rule!

Understanding Significant Figures

Before we tackle the problem, let's quickly recap what significant figures are. Significant figures are the digits in a number that are known with certainty plus one uncertain digit. They tell us how precisely a number is known. Here’s a quick rundown:

  1. Non-zero digits: All non-zero digits are significant. For example, in the number 345, all three digits are significant.
  2. Zeros between non-zero digits: Zeros located between non-zero digits are significant. For example, in the number 2007, all four digits are significant.
  3. Leading zeros: Leading zeros are not significant. For example, in the number 0.005, only the digit 5 is significant.
  4. Trailing zeros in a number containing a decimal point: Trailing zeros in a number containing a decimal point are significant. For example, in the number 3.50, all three digits are significant.
  5. Trailing zeros in a number not containing a decimal point: Trailing zeros in a number not containing a decimal point are not significant. For example, in the number 100, only the digit 1 is significant. However, if it's written as 100., then all three digits are significant.

Solving the Multiplication Problem

Okay, now let's get to the problem at hand: 5.12 x 6.802. The first thing we need to do is identify the number of significant figures in each value.

  • 5.12 has three significant figures.
  • 6.802 has four significant figures.

According to the rule, our final answer should have the same number of significant figures as the number with the fewest significant figures. In this case, that's three significant figures because 5.12 has only three.

Now, let's multiply the two numbers:

  1. 12 x 6.802 = 34.82624

We need to round this result to three significant figures. To do this, we look at the first four digits: 34.82. Since the next digit (2) is less than 5, we round down. Therefore, the answer to the correct number of significant figures is 34.8.

Analyzing the Options

Now let's look at the answer choices and see which one matches our result:

A. 34.82 B. 34.83 C. 34.8 D. 34.826

The correct answer is C. 34.8 because it has three significant figures, which is what we determined we needed. The other options have more than three significant figures, making them incorrect.

Why Significant Figures Matter

So, why do we even bother with significant figures? Well, imagine you're conducting an experiment to measure the density of a substance. You use a balance that measures to the nearest 0.01 gram and a graduated cylinder that measures to the nearest 1 mL. If you meticulously record your measurements and perform your calculations with the correct number of significant figures, you're ensuring that your final result accurately reflects the precision of your instruments. If you ignore significant figures, you might end up with a result that seems more precise than it actually is, which could lead to incorrect conclusions.

Moreover, significant figures play a crucial role in scientific communication. When scientists publish their findings, they include the appropriate number of significant figures to indicate the reliability of their data. This allows other scientists to evaluate the quality of the research and determine whether the results are trustworthy.

In fields like engineering and medicine, accuracy is paramount. Whether it's calculating the load-bearing capacity of a bridge or determining the correct dosage of a medication, precision can have life-or-death consequences. Therefore, understanding and applying the rules of significant figures is not just an academic exercise; it's a fundamental skill that can have real-world implications.

Common Mistakes to Avoid

Even though the concept of significant figures seems straightforward, there are several common mistakes that students make. Here are a few to watch out for:

  1. Forgetting the rules for zeros: As we discussed earlier, zeros can be tricky. Make sure you know the difference between leading, trailing, and captive zeros and when they are significant.
  2. Rounding too early: Always wait until the very end of your calculation to round. Rounding in the middle of a calculation can introduce errors and affect the accuracy of your final result.
  3. Ignoring significant figures in multi-step calculations: When you're performing a series of calculations, keep track of the number of significant figures at each step. Your final answer should have the same number of significant figures as the least precise measurement.
  4. Assuming all digits are significant: Just because a number has a lot of digits doesn't mean they're all significant. Remember to consider the precision of your measurements and instruments.

Practice Makes Perfect

The best way to master significant figures is to practice, practice, practice! Work through as many example problems as you can find, and don't be afraid to ask for help if you get stuck. Over time, you'll develop a feel for significant figures and be able to apply the rules automatically.

Conclusion

So, in conclusion, when multiplying 5.12 x 6.802 and considering significant figures, the correct answer is C. 34.8. Always remember to apply the rules of significant figures to ensure your calculations reflect the precision of your measurements. Keep practicing, and you'll become a pro in no time! You got this, future chemists!