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

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.

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.