The universal gas constant R is related to Avogadro's number and Boltzmann's constant by which equation?

• A. R = N_A / k_B
• B. R = N_A k_B
• C. R = k_B / N_A
• D. R = N_A m_B

Answer: (B)

Relating the energy scale per particle to the scale per mole. The universal gas constant comes from connecting Boltzmann's constant k_B, which is energy per degree of freedom per particle, with Avogadro's number N_A, which converts from particles to moles. From the kinetic view of an ideal gas, PV = N k_B T. For one mole of gas, N = n N_A, so PV = n (N_A k_B) T. This has the same form as PV = nRT, which means R = N_A k_B. Numerically, k_B ≈ 1.38×10^-23 J/K and N_A ≈ 6.022×10^23 1/mol, giving R ≈ 8.314 J/(mol·K). The other options don’t fit because they mix or invert the per-particle and per-mole relationships or use unrelated symbols, which would not yield the correct units or derivation.