[Paper Review] Measurement of the Boltzmann constant by Einstein. Problem of the 5-th Experimental Physics Olympiad. Sofia 9 December 2017

Todor M. Mishonov, Emil G. Petkov+6 more|arXiv (Cornell University)|Dec 29, 2017

Advanced Thermodynamics and Statistical Mechanics3 citations

TL;DR

This paper presents a hands-on experimental method for measuring the Boltzmann constant using thermal voltage fluctuations in a capacitor-resistor circuit, adapted from Einstein’s theoretical framework. Amplified voltage fluctuations are measured via a mean-square detector, and $k_B$ is extracted using the equipartition theorem, achieving a measured value of $1.182 \times 10^{-23}$ J/K with a 15% error relative to the accepted value, demonstrating a feasible educational application of fundamental statistical mechanics in high school and university-level physics labs.

ABSTRACT

Several consecutive experiments with specifically built set-up are described. Performing of the consecutive experimental tasks enables possibility to determine Boltzmann's constant $k_\mathrm{_B}$. The fluctuations of the voltage $U(t)$ of series of capacitors connected in parallel with a constant resistance are measured. The voltage is amplified 1~million times $Y=10^6$. The amplified voltage $YU(t)$ is applied to a device, which give the voltage mean quared in time $U_2=\left/U_0$. This voltage $U_2$ is measured with a multimeter. A series of measurements gives the possibility to determine the Boltzmann's constant from the equipartition theorem $C\left=k_\mathrm{_B}T$. In order to determine the set-up constant $U_0$ a series of problems connected with Ohm's law are given that are addressed to the senior students. For the junior high school students, the basic problem is to analyse the analog mean squaring. The students works are graded in four age groups S, M, L, XL. The last age group contains problems that are for university students (XL category) and include theoretical research of the set-up as an engineering device. This problem is given at the Fifth Experimental Physics Olympiad "Day of the Electron", on December 2017 in Sofia, organized by the Sofia Branch of the Union of Physicists in Bulgaria with the cooperation of the Physics Faculty of Sofia University and the Society of Physicists of the Republic of Macedonia, Strumica.

Motivation & Objective

• To provide a practical, low-cost experimental method for measuring the Boltzmann constant $k_B$ suitable for high school and university physics education.
• To adapt Einstein’s theoretical 1907 proposal for $k_B$ measurement into a reproducible, hands-on laboratory experiment using accessible electronic components.
• To enable students across multiple age groups to engage with fundamental statistical mechanics through a single, standardized experimental setup.
• To validate the method through real-world student experiments in the Fifth Experimental Physics Olympiad, assessing accuracy and educational impact.

Proposed method

• Measure voltage fluctuations $U(t)$ across a parallel combination of capacitors and a resistor at room temperature.
• Amplify the voltage signal by a factor of $Y = 10^6$ using a custom-built amplifier.
• Apply the amplified signal $YU(t)$ to a mean-square detector that outputs $U_2 = \langle (YU(t))^2 \rangle / U_0$, where $U_0$ is a calibration constant.
• Measure $U_2$ using a digital multimeter to obtain the mean-square voltage.
• Use the equipartition theorem $C \langle U^2 \rangle = k_B T$ to calculate $k_B$, where $C$ is the capacitance and $T$ is the absolute temperature.
• Determine the calibration constant $U_0$ through a series of Ohm’s law-based problems tailored for different student age groups.

Experimental results

Research questions

• RQ1 Can Einstein’s 1907 theoretical method for measuring the Boltzmann constant be practically realized using modern low-cost electronics in an educational setting?
• RQ2 To what extent can thermal voltage fluctuations in a simple RC circuit be used to extract $k_B$ with measurable accuracy in a high school or university laboratory?
• RQ3 How does the accuracy of the measurement depend on the precision of voltage measurements and temperature control?
• RQ4 Can a single experimental setup be effectively adapted to different educational levels—junior high, senior high, and university—through tiered problem sets?
• RQ5 What is the educational impact of using a real, reproducible experimental setup in a global physics olympiad context?

Key findings

• The experiment successfully measured the Boltzmann constant using thermal voltage fluctuations, yielding a value of $1.182 \times 10^{-23}$ J/K.
• The measured value deviates by approximately 15% from the accepted value of $1.38 \times 10^{-23}$ J/K, primarily due to imprecise measurement of the mean-square voltage $U_2$.
• Students in the Olympiad were able to reproduce the experiment and analyze data using standard tools like Excel, demonstrating feasibility in educational settings.
• The calibration constant $U_0$ was determined through a series of Ohm’s law-based tasks, enabling accurate extraction of $k_B$ from voltage measurements.
• The method was validated across four age groups (S, M, L, XL), with the XL category including theoretical analysis suitable for university-level students.
• The experimental setup was distributed to all participants and retained by schools post-Olympiad, enabling replication and long-term educational use.