Question

The gas constant is:

• A. ratio of Boltzmann constant and Avogadro's number
• B. product of Boltzmann constant and Avogadro's number
• C. ratio of Avogadro's number and Boltzmann constant
• D. product of square of Boltzmann constant and Avogadro's number

Hint:

Think of the constants this way: The universal gas constant \(R\) is for macroscopic amounts (per mole), while the Boltzmann constant \(k_B\) is for microscopic amounts (per molecule). Avogadro's number is the conversion factor between them.

Answer 1

The Correct Option is B

Step 1: State the two forms of the ideal gas law.- Molar form: \(PV = nRT\), where \(n\) is moles and \(R\) is the universal gas constant.- Molecular form: \(PV = Nk_BT\), where \(N\) is molecules and \(k_B\) is the Boltzmann constant.
Step 2: Establish the relationship between moles (\(n\)) and molecules (\(N\)).The number of molecules equals the number of moles multiplied by Avogadro's number (\(N_A\)), which is molecules per mole.\[ N = n N_A \]
Step 3: Set the two ideal gas law forms equal and solve for \(R\).\[ nRT = Nk_BT \]Substitute \(N = n N_A\):\[ nRT = (n N_A) k_B T \]Cancel \(n\) and \(T\) from both sides:\[ R = N_A k_B \]Therefore, the universal gas constant \(R\) is the product of Avogadro's number and the Boltzmann constant.