Problem 28 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

If the value of Avogadro number is \(6.023 \times 10^{23} \mathrm{~mol}^{-1}\) and the value of Boltzmann constant is \(1.380 \times 10^{-23} \mathrm{~J} \mathrm{~K}^{-1}\), then the number of significant digits in the calculated value of the universal gas constant is [Adv. 2014]

Step-by-Step Solution

Answer

The universal gas constant has 4 significant digits.

Step 1: Write Formula for Universal Gas Constant

The universal gas constant \( R \) is given by the product of Avogadro's number \( N_A \) and the Boltzmann constant \( k \). The formula is: \[ R = N_A \times k \]

Step 2: Substitute Known Values

Substitute the given values into the formula. \( N_A = 6.023 \times 10^{23} \), and \( k = 1.380 \times 10^{-23} \). So, \[ R = (6.023 \times 10^{23}) \times (1.380 \times 10^{-23}) \]

Step 3: Calculate the Product

To calculate \( R \), multiply the numbers \( 6.023 \) and \( 1.380 \), and add the exponents \( 23 + (-23) = 0 \). The numerical multiplication is \( 6.023 \times 1.380 = 8.31474 \), so: \[ R = 8.31474 \times 10^{0} = 8.31474 \]

Step 4: Determine Significant Digits

The number with the least number of significant digits is \( 1.380 \) which has 4 significant digits. Therefore, the calculated value of \( R \) should also have 4 significant digits. Round the result to get \( 8.315 \).

Key Concepts

Significant Figures

Significant figures are essential in scientific calculations because they indicate the precision of a measurement. In any measured or calculated quantity, significant figures are all the digits known with certainty, plus one that is estimated. For instance, when you are given a number like 6.023, it possesses four significant figures (6, 0, 2, and 3). This means it was measured with a certain level of precision.

In a calculation, the number of significant figures in the result is determined by the initial number in a computation with the fewest significant figures. For example, in the product of two numbers, like in the exercise, the number with the least significant figures guides how you should round the result. This ensures that your answer is not falsely more precise than the least precise measurement used to derive it.

Understanding significant figures helps in imparting precision to the results, maintaining consistency across measurements, and eliminating the uncertainty of excessive figures. It's vital to round off appropriately to reflect the true precision of the measurement or calculation.

Avogadro's Number

Avogadro's number, denoted as \( N_A \), is an incredibly important constant in chemistry and physics. It represents the number of constituent particles, usually atoms or molecules, contained in one mole of a substance. This number is universally recognized as approximately \( 6.022 \times 10^{23} \), but for different contexts, it might be presented as \( 6.023 \times 10^{23} \), as seen in certain scientific exercises.

Avogadro's number plays a critical role in connecting the macroscopic and microscopic worlds. It allows chemists to count entities at a molecular level by weighing them. This helps in calculating amounts of substances needed for reactions, figuring out molecular weights, and many other tasks. Without \( N_A \), it would be nearly impossible to deal with the vast number of atoms or molecules involved in practical chemical reactions.

Therefore, its usage extends beyond academic exercises; it is foundational to practical laboratory work and theoretical science, bridging the gap between measuring instruments and atomic-scale reactions.

Boltzmann Constant

The Boltzmann constant, denoted as \( k \), is a fundamental constant in physics that provides a bridge between macroscopic and microscopic physical quantities. It is defined as \( 1.380 \times 10^{-23} \, \mathrm{J} \, \mathrm{K}^{-1} \), which underlines its role in connecting temperature, measured in kelvins, to energy, measured in joules.

In the realm of statistical mechanics, the Boltzmann constant is critical for explaining thermal behavior at microscopic scales. It helps define the average kinetic energy of particles in a gas, linking it to the temperature of the gas. This relationship forms the backbone of the kinetic theory of gases.

In calculations like those for the universal gas constant, \( k \) is multiplied by Avogadro's number to relate molecular motion to measurable energy changes. Boltzmann constant's universality ensures consistency across these varied physical phenomena, supporting predictive models in thermal dynamics and providing insights in fields ranging from thermodynamics to cosmic background radiation measurement.