Question

If the value of \(\Lambda\) vogadro Number is \(6.023 \times 10^{23} \mathrm{~mol}^{1}\) and the value of Boltamann constant is \(1.380 \times 10^{-23} J K^{1}\), then the number of significant digits in the calculated value of the universal gas oonstant is (The answer to the question is a single-digit integer, ranging from 0 to 9 )

Short answer

4

Step-by-step solution

Understanding Significant Figures

Significant figures in a measured or calculated quantity are the digits that carry meaning contributing to its precision. This includes all the non-zero digits, any zeros between them, and any trailing zeros in the decimal part. The number of significant figures is determined by the least precise measurement used in the calculation.

Identify Significant Figures in the Avogadro's Number

Avogadro's Number, \(6.023 \times 10^{23} \mathrm{~mol}^{-1}\), has four significant figures because it includes all non-zero digits and any zeros between them.

Identify Significant Figures in the Boltzmann's Constant

Boltzmann's Constant, \(1.380 \times 10^{-23} J K^{-1}\), has four significant figures, as it has three non-zero digits and one leading zero after the decimal point that is between non-zero digits.

Calculating the Universal Gas Constant

The universal gas constant (R) is calculated using the formula \(R = \Lambda \times k\), where \(\Lambda\) is Avogadro's Number and \(k\) is Boltzmann's Constant. Since both constants have four significant figures, the resulting value of R will have four significant digits because the number of significant figures in the result is determined by the number with the fewest significant figures used in the calculation.

Conclusion

The calculated value of the universal gas constant will have four significant figures, which is the same number of significant digits present in both the Avogadro's Number and the Boltzmann's Constant.

Key concepts

Understanding Significant Figures

The concept of significant figures is central to precision in scientific calculations. It's essential to understand that significant figures include all not-zero digits, zeros between those digits, and trailing zeros in a decimal number that represent measured or estimated precision. They impact calculations by setting the standard for how precise the final answer should be.

For example, in the universal gas constant calculation, determining the significant figures involves looking at both Avogadro's Number and Boltzmann's Constant to understand their precision levels. From there, the calculation's precision is constrained by the number with the least number of significant figures. Typically, when performing division or multiplication, the final result should be reported with the same number of significant figures as the value with the fewest significant digits.

Avogadro's Number

Avogadro's Number, which is approximately equal to \(6.023 \times 10^{23} \mathrm{\ mol^{-1}}\), represents the number of constituent particles (usually atoms or molecules) in one mole of a substance. It bridges the macroscopic and microscopic worlds, allowing chemists and physicists to work with amounts of substances in a meaningful way.

This constant not only underpins important concepts in chemistry and physics but also plays a crucial role in calculating other constants like the universal gas constant. With its four significant figures, Avogadro's Number ensures that expressions of mole quantities are communicated with the proper precision to be universally understood and applied in scientific communities.

Boltzmann's Constant

Boltzmann's Constant, represented as \(1.380 \times 10^{-23} J K^{-1}\), is a fundamental physical constant that relates the average kinetic energy of particles in a gas with the temperature of the gas. This significant figure also comprises four significant digits, which entails that it reflects the precision with which this constant is known.

It serves as a conversion factor between temperature on the microscopic scale (measured in energy units) and the macroscopic scale (measured in degrees). Understanding Boltzmann's Constant is essential for students diving into statistical mechanics and thermodynamics, as it is instrumental in the calculation of the universal gas constant, and thereby, understanding of gas laws and the behavior of ideal gas.