Electrical Equivalent of Heat theory

James Prescott Joule's classic experiments between 1840 and 1843 established that electrical energy is not a separate imponderable caloric fluid, but a quantitative manifestation of universal mechanical energy. Joule's First Law of Electrical Heating states that when a steady direct current I flows through a conductor of resistance R across potential difference V for duration t, the electrical work W delivered by the electric field is completely converted into internal thermal energy at the rate P = dW/dt = V · I = I² R = V² / R. In a resistive medium, conduction electrons accelerated by the electric field collide inelastically with lattice ions (in solids) or polar water molecules (in liquids), dissipating their kinetic energy into randomized thermal vibrations (phonons). By the First Law of Thermodynamics, in an adiabatic system the total mechanical/electrical energy expended W must equal the heat gained Q multiplied by a universal dimensional proportionality constant J: W = J · Q. In SI units, both work and heat are measured in Joules (making the ratio exactly 1.000 J/J), but in historical and engineering units where heat is quantified in calories (the thermal energy required to raise 1 gram of water by 1 °C), J represents the fundamental conversion factor: J ≈ 4.1860 J/cal.

The calorie was historically defined as the quantity of heat required to raise the temperature of one gram of air-free water by one degree Celsius under standard atmospheric pressure (101.325 kPa). However, because the specific heat capacity of liquid water c_w(T) varies slightly with temperature (exhibiting a minimum of 4.178 J/g °C at 35 °C and rising to 4.218 J/g °C near 0 °C and 100 °C), several precise definitions emerged: (1) The 15 °C calorie (cal₁₅): The heat required to raise 1 g of water from 14.5 °C to 15.5 °C, experimentally measured as 4.1855 J. (2) The Thermochemical Calorie (cal_th): Formally defined by the National Bureau of Standards in 1935 as exactly 4.184 J. (3) The International Steam Table Calorie (cal_IT): Defined by the Fifth International Conference on Properties of Steam (London, 1956) as exactly 4.1868 J (1/860 of an international watt-hour). The value determined in this laboratory (J = 4.1860 J/cal) corresponds to the mean laboratory calorie across the 15 °C to 40 °C working range, directly matching standard NIST thermodynamic tables.

In any quantitative calorimetry experiment, the heat produced by the immersion heater is shared between the liquid water and the surrounding apparatus components in contact with it: the inner copper vessel, the brass heater sheath, the glass thermometer bulb, and the motorized stirrer paddle. The total heat capacity of the system C_sys (in cal/°C) is the sum of these contributions: C_sys = m_w c_w + m_vessel c_Cu + C_accessories. To simplify calculations, physicists define the 'water equivalent' (W_eq) of the calorimeter as the mass of water that would absorb the exact same quantity of heat as the metal vessel and accessories for the same temperature rise: W_eq = (m_vessel c_Cu + C_accessories) / c_w. For our 100.0 g copper vessel (specific heat c_Cu = 0.093 cal/g °C) and immersion accessories, W_eq = 9.3 g of water. Adding this to the 250.0 g of distilled water gives an effective system heat capacity of C_sys = (250.0 + 9.3) × 1.000 = 259.3 cal/°C. Neglecting the water equivalent would cause an immediate systematic underestimation of heat absorbed by 3.6%, distorting the calculated value of J.

Because the calorimeter vessel operates at a temperature different from the ambient room, heat exchange inevitably occurs across the boundary via thermal conduction, convection, and radiation. According to Newton's Law of Cooling, the rate of heat loss to the surroundings is proportional to the temperature excess over the environment: dQ_loss / dt = k · A · (T − T_ambient). If uncorrected, this heat loss causes the final temperature T to be lower than theoretical, making Q appear too small and inflating the calculated J. To eliminate this systematic error without tedious numerical integration, Count Rumford developed the Method of Equal Temperature Differentials (Rumford compensation). The water is initially cooled so that its starting temperature T₀ is exactly as far below room temperature as its expected final temperature T_f is above room temperature: T_ambient − T₀ = T_f − T_ambient = ΔT / 2. During the first half of the heating run, heat flows from the warmer room into the colder calorimeter; during the second half, heat flows from the hotter calorimeter into the room. Because the heating curve is nearly linear, the two integrated heat exchanges cancel out almost perfectly: ∮ dQ_ambient dt ≈ 0.

Plotting total electrical energy W = V · I · t (in Joules) against the heat absorbed Q = C_sys · ΔT (in calories) transforms the data into a standard linear model: W = J · Q + b, where independent variable x = Q (in cal), dependent variable y = W (in J), slope m = J (in J/cal), and intercept b is ideally 0.0 J. Ordinary least-squares regression across six distinct heating durations (120 s to 720 s) averages out thermometer reading quantization (±0.02 °C) and timer switching latency. The slope directly yields the electrical equivalent of heat J_est. The reciprocal of the slope represents the heat equivalent of electrical energy: a = 1 / J (in calories per Joule, a ≈ 0.2389 cal/J), which represents the exact quantity of thermal heat generated by one Joule of electrical work. Furthermore, the vertical intercept b serves as a diagnostic indicator: a positive intercept indicates that a constant amount of electrical energy was consumed in heating the vessel wall before the thermometer began to respond, while a negative intercept reflects initial heat absorbed from the room.

A comparison between mechanical friction calorimetry (such as the Callendar-Rowland rotating drum method used in Lab 29) and electrical immersion calorimetry highlights major advances in experimental metrology. In mechanical calorimetry, work is performed against mechanical friction: W = 2π r M g N. Mechanical methods suffer from several intractable systematic errors: mechanical wear of the friction belt, audible acoustic noise radiation, air turbulence, and non-uniform frictional heat generation across the drum surface. In contrast, electrical calorimetry eliminates all mechanical moving contacts: electrical power is delivered directly inside the fluid volume by a stationary resistance coil and can be measured with digital voltmeters and ammeters with extraordinary accuracy (< 0.05%). Furthermore, electrical heating allows instantaneous on-off switching, constant power regulation via electronic supplies, and complete immersion of the heating source, ensuring that nearly 100% of the dissipated energy enters the liquid.

In high-precision electrical calorimetry, two subtle sources of systematic error must be accounted for: lead wire resistance and mechanical stirrer power dissipation. (1) Lead Resistance: The connecting wires between the power supply and calorimeter possess finite resistance (R_lead ≈ 0.05 Ω). If voltage is measured at the power supply terminals, the calculated energy includes heat dissipated in the external room wiring. To eliminate this error, voltmeter leads must be connected directly to the calorimeter binding posts (four-terminal Kelvin sensing), ensuring that only the voltage drop across the submerged heater is integrated. (2) Stirrer Dissipation: Mechanical stirring rotates a paddle through viscous water, doing fluid work that is converted by viscous shear into heat: P_stirrer = τ · ω. If stirring is too vigorous (e.g., high-speed motorized blade), stirrer work can add 0.5 to 1.0 mW, inflating the measured temperature rise. By using a low-speed (60 rpm) spiral stirrer and non-conducting shaft, stirrer power is kept strictly below 0.02 mW, which contributes less than 0.005 °C over a 10-minute trial.