Animated Solution for Chemistry - Basic Concepts in Chemistry: If the value of Avogadro number is $6.023\times 10^{23}{\text{ mol}}^{-1}$ and the value of Boltzmann constant is $1.380\times 10^{-23}{\text{ J K}}^{-1}$, then the number of significant digits in the calculated value of the universal gas constant is

The problem asks us to determine the number of significant digits in the calculated value of the universal gas constant, $R$, given the values of the Boltzmann constant, $k$, and Avogadro's number, $N_A$.

In physical chemistry and thermodynamics, the universal gas constant $R$ acts as a macroscopic bridge to the microscopic world described by the Boltzmann constant $k$. The relationship is elegantly simple:

where $N_A$ is Avogadro's number, representing the number of particles in one mole.

We are given: - $k=1.380\times 10^{-23}{\text{ J K}}^{-1}$ - $N_A=6.023\times 10^{23}{\text{ mol}}^{-1}$

Let's substitute these into our equation:

When multiplying these terms, the powers of 10 cancel out perfectly ($10^{-23}\times 10^{23}=10^0=1$). We are left with the multiplication of the decimal parts:

In science, a calculated result cannot be more precise than the least precise measurement used in the calculation. The rule for multiplication and division states that the final answer must have the same number of significant figures as the measurement with the fewest significant figures.

Let's analyze our given values: - $1.380$ has 4 significant figures (the trailing zero after the decimal point is significant). - $6.023$ has 4 significant figures.

Since both values have exactly 4 significant figures, our final calculated value for $R$ must also be rounded to 4 significant figures.

Rounding $8.31174$ to 4 significant figures gives:

The question specifically asks for the number of significant digits in this calculated value.

Therefore, the number of significant digits is 4.