Problem Statement
Given $R = 8.314$ J/mol·K and $N_A = 6.022 \times 10^{23}$ mol$^{-1}$, calculate Boltzmann’s constant $k_B$. Also find the average kinetic energy of a gas molecule at 27°C.
Given Information
- All quantities, constants, and constraints stated in the problem above
- Physical constants used as needed (see Concepts section)
Physical Concepts & Formulas
This problem draws on fundamental physical principles. The key is to identify which conservation law or field equation governs the system, then apply it systematically. Dimensional analysis can always be used to verify that the final answer has the correct units. Working from first principles — rather than memorising formulas — builds deeper understanding and allows tackling novel problems.
- Identify the relevant physical law (Newton’s laws, conservation of energy/momentum, Maxwell’s equations, etc.)
- State the mathematical form of that law as it applies here
- Check dimensions at every step: both sides of an equation must have the same units
Step-by-Step Solution
Problem Statement
Given $R = 8.314$ J/mol·K and $N_A = 6.022 \times 10^{23}$ mol$^{-1}$, calculate Boltzmann’s constant $k_B$. Also find the average kinetic energy of a gas molecule at 27°C.
Given Information
- $R = 8.314$ J/mol·K
- $N_A = 6.022 \times 10^{23}$ mol$^{-1}$
- $T = 27°C = 300$ K
Physical Concepts & Formulas
$$k_B = \frac{R}{N_A}$$
$$\langle KE \rangle_{translation} = \frac{3}{2}k_B T$$
Step-by-Step Solution
Step 1: Calculate $k_B$.
$$k_B = \frac{8.314}{6.022 \times 10^{23}} = 1.381 \times 10^{-23} \text{ J/K}$$
Step 2: Average kinetic energy at 300 K.
$$\langle KE \rangle = \frac{3}{2} \times 1.381 \times 10^{-23} \times 300$$
Step 3: Compute.
$$= \frac{3}{2} \times 4.143 \times 10^{-21} = 6.214 \times 10^{-21} \text{ J}$$
Worked Calculation
$$k_B = \frac{R}{N_A} = \frac{8.314}{6.022 \times 10^{23}} = 1.381 \times 10^{-23} \text{ J/K}$$
$$\langle KE \rangle = \frac{3}{2}k_BT = 6.21 \times 10^{-21} \text{ J}$$
Answer
$k_B = 1.381 \times 10^{-23}$ J/K; $\langle KE \rangle = 6.21 \times 10^{-21}$ J
Physical Interpretation
Boltzmann’s constant is the bridge between macroscopic thermodynamics (temperature, in Kelvin) and microscopic statistical mechanics (energy per molecule). The 2019 SI redefinition fixed $k_B$ exactly as $1.380649 \times 10^{-23}$ J/K, thereby redefining the Kelvin in terms of this fundamental constant rather than the triple point of water.
Worked Calculation
Substituting all given numerical values with their units into the derived formula:
$$\text{Numerical result} = \text{given expression substituted with values}$$
Answer
$$\boxed{\langle KE \rangle = \frac{3}{2}k_BT = 6.21 \times 10^{-21} \text{ J}}$$
Physical Interpretation
The answer should be checked for dimensional consistency and physical reasonableness: is the magnitude in the expected range for this type of problem? Does the answer change in the correct direction when parameters are varied (e.g., increasing mass should increase momentum, increasing distance should decrease field strength)? These sanity checks are as important as the calculation itself.
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