Significant Digits in Universal Gas Constant Calculation

To find the number of significant figures in the computed value of the universal gas constant ($R$), we use the relationship between the universal gas constant, Avogadro's number ($N_A$), and the Boltzmann constant ($k_B$):

$R = N_A \times k_B$

Given the values from the problem:

1. Avogadro's number, $N_A = 6.023 \times 10^{23} \text{ mol}^{-1}$

   • Number of significant figures in $N_A = 4$ (the digits are 6, 0, 2, and 3).

2. Boltzmann constant, $k_B = 1.380 \times 10^{-23} \text{ J K}^{-1}$

   • Number of significant figures in $k_B = 4$ (the digits are 1, 3, 8, and 0).

According to the rules for significant figures in multiplication, the final result must retain the same number of significant figures as the term with the fewest significant figures.

Calculating $R$: $R = (6.023 \times 10^{23}) \times (1.380 \times 10^{-23}) \text{ J K}^{-1} \text{mol}^{-1}$ $R = 6.023 \times 1.380 = 8.31174 \text{ J K}^{-1} \text{mol}^{-1}$

Rounding $8.31174$ to $4$ significant figures gives: $R \approx 8.312 \text{ J K}^{-1} \text{mol}^{-1}$

Thus, the number of significant figures in the calculated value of the universal gas constant is $4$.