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If the value of Avogadro number is $6.023\times { 10 }^{ 23 }{ mol }^{ -1 }$ and the value of Boltzmann constant is $1.380\times { 10 }^{ -23 }{J}{ K }^{ -1 }$, then the number of significant figures in the calculated value of Universal gas constant is:
A) 4
B) 5
C) 3
D) 2

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Last updated date: 24th May 2024
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Answer
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Hint: We know that the number of significant figures in the result of any mathematical operation of two numbers will be the least number of significant figures of both numbers. Try to find out the formula for gas constant and calculate its value in the required number of significant figures.

Complete step by step answer:
We know that the Kinetic energy of one mole of an ideal gas is $\dfrac { 3 }{ 2 }$RT
Where R is Universal gas constant and T is the temperature of the gas.
We know that the kinetic energy of a single atom of that ideal gas is $\dfrac { 3 }{ 2 }$kT
Where k is Boltzmann constant

1 mole of gas contains an Avogadro number of atoms. For Ideal gas, there will be no atom to atom interactions. The kinetic energy of Avogadro number of atoms is N×3/2 kT which is equal to the kinetic energy of one mole. Avogadro number N = $6.023\times { 10 }^{ 23 }{ mol }^{ -1 }$

$\dfrac { 3 }{ 2 }$RT = $\dfrac { 3 }{ 2 }$NkT

R = Nk
Boltzmann constant k = $1.380\times { 10 }^{ -23 }{J}{ K }^{ -1 }$

R = $6.023\times { 10 }^{ 23 }\times 1.380\times { 10 }^{ -23 }{ J }{ K }^{ -1 }{ mol }^{ -1 }$

R = $6.023\times 1.380{ J }{ K }^{ -1 }{ mol }^{ -1 }$
Number of significant digits in Avogadro number will be 4 those are $6.023\times { 10 }^{ 23 }{ mol }^{ -1 }$
Number of significant digits in Boltzmann constant will be 4 those are $1.380\times { 10 }^{ -23 }{J}{ K }^{ -1 }$
The number of significant figures present in the Avogadro number and Boltzmann constant are the same. So Universal gas constant will contain the same number of significant figures which is equal to 4.
R = $8.31174 { J }{ K }^{ -1 }{ mol }^{ -1 }$ which is rounded to 4 significant figures and rounded value is R = $8.312{ J }{ K }^{ -1 }{ mol }^{ -1 }$

Therefore, option A is the correct answer.

Note: While counting the number of significant figures we may make a mistake so we need to follow all the rules of significant figures correctly. We observe that the number of significant figures in the Gas constant and Boltzmann constant are the same.
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