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: Recall the two forms of the ideal gas law. - The molar form is \(PV = nRT\), where \(n\) is the number of moles and \(R\) is the universal gas constant. - The molecular form is \(PV = Nk_BT\), where \(N\) is the number of molecules and \(k_B\) is the Boltzmann constant.
Step 2: Relate the number of moles (\(n\)) to the number of molecules (\(N\)). The number of molecules is equal to the number of moles multiplied by Avogadro's number (\(N_A\)), which is the number of molecules per mole. \[ N = n N_A \]
Step 3: Equate the two forms of the ideal gas law 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: \[ R = N_A k_B \] Thus, the universal gas constant \(R\) is the product of Avogadro's number and the Boltzmann constant.