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Heat-engine calorimetry measures heat transfers so that observed work can be compared with the correct energy input. Measuring motion alone cannot supply the denominator of thermal efficiency.

The institutional Laboratory 16: Stirling Engine describes a modern teaching arrangement: an ethanol lamp drives a Stirling engine coupled to an electrical output device, which supplies a resistor immersed in water in a calorimeter. The prescribed readouts include fuel mass consumed, the pressure-volume loop, electrical readings, and water temperature. The sequence separates fuel input, gas-cycle work, electrical output, and Joule heating of the water. Warming of the water is the final useful output; it is not the engine's rejected $Q_c$. The handout distinguishes warm-up from steady operation and asks students to compare the final warming with direct heating by the lamp. It gives a procedure, not a table of measured efficiencies. Its pp. 1–3 therefore establish an apparatus and measurement design without supplying a new experimental result.

For a proposed calibration, use a known input of electrical work to determine the effective heat capacity $C_{\mathrm{eff}}$ of the water, vessel and immersed parts; stir to make the measured temperature representative. A matched run with no input estimates exchange with the room. For constant $C_{\mathrm{eff}}$ and no phase transition, the resulting model iswhere $Q_{\mathrm{leak},\mathrm{out}}$ is the signed net heat lost to the room during the measurement. Here $E_{\mathrm{in}}$ is supplied electrical work when the resistor lies inside the calibration boundary; for a separate external heat input it is the received heat. These are proposed controls, not controls reported in that handout. Slow cooling, changing material properties and imperfectly matched room exchange limit this inference; the approach to thermal equilibrium and heat capacity matter as much as the thermometer reading.

To measure the actual engine-input $Q_h$, one must instrument its hot-side transfer separately, or close a calibrated energy balance around a specified heater boundary. A heater's electrical work includes changes in its stored internal energy and heat transfer that bypasses the engine. At periodic operation a cold-side flowing-water measurement can instead estimate ${\dot{Q}}_c=\dot{m}{c}_p(T_{\mathrm{out}}-T_{\mathrm{in}})$, if mass flow rate $\dot{m}$ and specific heat capacity at constant pressure $c_p$ are known, with negligible storage and other heat exchange in that measurement boundary. It measures only the rejection captured by that stream. Include all remaining transfers before using the residual to infer ${\dot{Q}}_h=P_{\mathrm{out}}+{\dot{Q}}_c$. This measurement discipline makes a comparison with Carnot's theorem meaningful.

A heat exchanger transfers heat between a fluid and another fluid or its surroundings. In measuring the Joule-Thomson effect with a porous plug, an upstream coil restores the inlet temperature before the measurement throttle; it belongs outside the approximately insulated measurement control volume.

On $0<x<L$ with zero-temperature endpoints, let the initial temperature rise linearly from zero to height $H$ at $L/2$ and then fall linearly to zero. Integration by parts on the two halves gives Fourier sine series coefficients $8H\sin (n\pi /2)/({\pi }^2{n}^2)$. Under the heat equation $T_t=D{T}_{xx}$ with $D>0$, each coefficient decays by $e^{-D(n\pi /L{)}^{2}t}$, giving the displayed solution. Only odd spatial modes occur; absolute coefficient summability proves uniform convergence for $t\ge 0$ via the Weierstrass M-test. This spatial profile also occurs in a midpoint-plucked fixed-end string, where the time factors oscillate instead of decaying.

On a bar of length $L$, let the initial temperature increase linearly from zero at either endpoint to height $H$ at the midpoint. With thermal diffusivity $D>0$ and insulated Neumann boundary conditions, a Fourier cosine series gives the displayed solution. Its constant term is the initial integral average $H/2$. Integration by parts on the two straight segments gives zero coefficients for odd modes and modes divisible by four; the remaining coefficients are $-4H/[{\pi }^2(2j+1{)}^2]$. Nonconstant modes decay, leaving the uniform temperature $H/2$, consistent with conservation of the total heat. This contrasts with the zero-temperature boundary solution in heat flow from a triangular initial temperature, where heat can leave the bar and the limit is zero. The initial endpoint slopes do not satisfy the insulated condition, so derivatives need not converge to their initial endpoint values as $t\downarrow 0$; the series satisfies the Neumann condition for every $t>0$ and recovers the continuous initial profile.

A negative-temperature system is hotter than every positive-temperature system. If heat $\delta Q$ passes from it to a positive-temperature body, the total entropy change is

An instrument that combines subsurface temperature measurements with constraints on thermal conductivity to estimate a local heat flux. Multiple sensor depths help separate a stable geothermal gradient from surface-temperature transients and installation disturbance. Conductive heat flow can differ from total heat transport where fluid circulation carries substantial energy.

For a quasistatic simple compressible system with only pressure-volume work, the direction of heat transfer is determined by $dU+pdV$. A reversal occurs where this vanishes and changes sign. It need not coincide with a maximum or minimum of temperature, because the substance can still do work when its internal energy is stationary.

Thermal energy generated per unit time, often specified per volume as $q^{\prime \prime \prime }$ in $\mathrm{W}{\mathrm{m}}^{-3}$. Integrating a volumetric source over a region gives its total thermal power.

A useful optically thin model subtracts volumetric heating from radiative cooling. At equilibrium its net cooling vanishes. Pressure-confined temperature perturbations are governed by the derivative of net cooling per unit mass at fixed pressure, so a negative derivative is unstable. The heating law, species abundances and radiation field must be specified before predicting an equilibrium curve. thermal conduction and finite sound-crossing times limit the isobaric approximation.

A heating curve records temperature against accumulated supplied heat under a specified constraint. Within one thermodynamic phase, its slope is $dT/dQ=1/C_X$, where $C_X$ is the constrained heat capacity. A latent heat plateau instead records changing phase fraction at the phase coexistence temperature. The sensible heat example combines two sloping intervals and a melting plateau in one heating budget.

A heat pump uses work to deliver heat to the region we want to keep warm. Its useful output is the hot-side delivery $Q_h$. The same operating machine can be described as a refrigerator when the cold-side extraction $Q_c$ is the objective: which output is useful changes the coefficient of performance, while the energy balance stays $Q_h=Q_c+W$. Heating a water bath by taking heat from a colder water bath makes this distinction visible in the heat-pump calorimetry experiment below. ^[ref]

In the illustrative budget from refrigerator, supplying $1\mathrm{kJ}$ of work delivers $4\mathrm{kJ}$ of useful heat, giving a heating coefficient of performance of $4$. The additional $3\mathrm{kJ}$ came from the colder region. Compare an ideal electrical heater whose entire input work becomes heat in the target region: it delivers $1\mathrm{kJ}$ for that input. The appeal of a heat pump is this ability to move existing energy, while the Second law of thermodynamics limits the work saving for a given pair of temperatures.

A motivating bench experiment asks whether a hot bath receives more heat than the electrical work supplied to a heat pump. PHYWE's authored operating instructions, section 4, describe a sealed vapor-compression refrigeration apparatus with its cold and hot coils immersed in separate water vessels. The quantitative procedure calls for insulated vessels, stirring to reduce temperature gradients, timed temperature readings, and a wattmeter for the input. It waits about five minutes for operating conditions to settle before measurements and explicitly includes the glass vessel and immersed copper pipe when calculating the heated body's heat capacity. One bath cooling while the other warms makes the origin of the delivered energy observable. ^[ref]

The following analysis extends that procedure to simultaneous measurements of both finite baths. Weigh the water in each vessel; determine the effective heat capacities $C_c$ and $C_h$, including immersed hardware, by a separate known electrical-input calibration over the working temperature range. A matched pair of calibrated thermometers placed in stirred water measures bulk $T_c(t)$ and $T_h(t)$; keeping probes away from the coils avoids mistaking a local contact temperature for the whole bath. Record the room temperature and synchronise the electrical meter and thermometry. Use short intervals after startup, with no boiling or freezing and small enough temperature changes that constant effective heat capacities are a reasonable approximation. These are proposed measurement controls, rather than reported results from the manual.

Over a common interval $[t_0,t_1]$, the uncorrected bath estimates and measured electrical input areHere the wattmeter records real electrical power $P_{\mathrm{el}}$. The integral handles changing input, and the signs make both estimated useful heat quantities positive during heating and cooling. The effective heat capacity includes water mass times its specific heat capacity plus the response of the vessel and immersed hardware. Without that correction, equal temperature changes need not imply equal heat transfers.

Insulation reduces heat leaks but does not establish an isolated system. A warmer bath can lose heat to the room while a colder bath gains it. As a proposed control, measure pump-off drift at matched bath and room temperatures, with the same stirring and insulation. Writing $L_h>0$ for outward hot-bath leak power and $L_c>0$ for inward cold-bath leak power, the bath balances giveIntegrate these rates over the same interval before dividing by $W_{\mathrm{el}}$. For example, in a pump-off hot-bath control, $L_h=-C_{h}d{T}_h/dt$; in the cold-bath control, $L_c=C_{c}d{T}_c/dt$. Check that a stopped machine does not create a different leak path from the running one; otherwise use an independent bath calibration and keep that discrepancy as an uncertainty. Repeat at the same starting temperatures to distinguish drift from a change in COP as the baths separate.

In the illustrated toy model, $C_h=C_c=4500\mathrm{J}{\mathrm{K}}^{-1}$, $P_{\mathrm{el}}=50\mathrm{W}$, ${\dot{Q}}_h=150\mathrm{W}$, ${\dot{Q}}_c=100\mathrm{W}$, $L_h=15\mathrm{W}$, and $L_c=5\mathrm{W}$ are stipulated constants for $180\mathrm{s}$. The slopes are $d{T}_h/dt=0.030\mathrm{K}{\mathrm{s}}^{-1}$ and $d{T}_c/dt=-19/900\mathrm{K}{\mathrm{s}}^{-1}$. Consequently,The raw heating and cooling ratios are $2.7$ and $1.9$, whose difference is $0.8$. Correcting the bath leaks gives $3$ and $2$, whose difference is one. The apparent failure of $Q_h=Q_c+W$ was incomplete measurement of heat, not a violation of the first law of thermodynamics.

Real apparatus can also release motor heat and pipe heat directly to the room, and store internal energy during startup. For a machine boundary that receives electrical work, an additional outward heat transfer $Q_{\mathrm{room}}$ gives $Q_h+Q_{\mathrm{room}}=Q_c+W_{\mathrm{el}}-\Delta U_{\mathrm{machine}}$. The residual $Q_h-Q_c-W_{\mathrm{el}}$ therefore tests the boundary and corrections; do not force it to zero. Include any auxiliary electrical work in the input if quoting whole-apparatus COP, and account separately for stirring work deposited in a bath. Ignoring vessel heat capacity, sensor lag or room leaks can bias the inferred result. These limitations follow from extending the measured energy budget, and the manual's explicit hardware correction illustrates why such boundaries matter.

Finally, these vessels are finite bodies whose temperatures change; they are not fixed thermal reservoirs. Their ratio of integrated useful heat to integrated input work is an interval COP. A Carnot's theorem comparison using simultaneous bath temperatures is a local approximation only when the machine settles much faster than the baths change. An exact reversible comparison for changing bath temperatures instead uses the bath entropy changes, for example $C_h\ln (T_{h,1}/T_{h,0})+C_c\ln (T_{c,1}/T_{c,0})$ for constant heat capacities, negligible net machine entropy change and no room leaks. The experiment's beauty is that two thermometers, known heat capacities and an electrical-energy measurement expose both conservation of energy and the practical gap from reversibility.

As $t\to 0$, the conditional mean rotational energies approach $U_{\mathrm{p}}/N=0$ and $U_{\mathrm{o}}/N=2k_{\mathrm{B}}{\theta }_r$, using the para $J=0$ level as the common zero. A frozen mixture with ortho fraction $x$ therefore retains $U_{\mathrm{fr}}/N\to 2x{k}_{\mathrm{B}}{\theta }_r$. If a bath keeps the sample at low temperature while a catalyst permits conversion to almost pure parahydrogen, the rotational heat delivered to the bath tends to $Q_{\mathrm{out}}/N=2x{k}_{\mathrm{B}}{\theta }_r$, or $3k_{\mathrm{B}}{\theta }_r/2$ for normal hydrogen.

One may shift the ortho ladder to its own lowest level for computing its conditional heat capacity, because a constant energy offset disappears on differentiation at fixed composition. That offset must be restored before comparing species or calculating conversion heat. Otherwise the calculation loses precisely the stored rotational energy that conversion releases. The expression above accounts for the internal rotor contribution at fixed bath temperature; it does not model liquid intermolecular energies, latent heat, or an adiabatic temperature rise.

A catalyst changes the conversion rate and allows heat removal during cooling; it does not change the free-gas equilibrium weights at a specified temperature. This connects the microscopic exchange rule to refrigeration: delayed conversion supplies a continuing internal heat source. Rotational Raman spectroscopy offers a complementary population probe because its $\Delta J=\pm 2$ transitions stay within the even or odd ladder; line intensities require the appropriate species populations and transition response, not a bare $3:1$ intensity rule.

Heat transfer asks how quickly heat crosses a boundary and how the temperature varies along its path. Its principal mechanisms are thermal conduction, convection and thermal radiation. A warm wall can conduct heat through its interior while exchanging heat with moving air and emitting thermal radiation at its surfaces. Identifying these paths lets us estimate heating requirements and cooling times.

Distinguish the accumulated heat $Q$ from its rate $\dot{Q}=dQ/dt$, a power measured in $\mathrm{W}$. The heat flux $q$ is the rate per unit area, measured in $\mathrm{W}{\mathrm{m}}^{-2}$: a uniform normal heat flux across area $A$ gives $\dot{Q}=Aq$. The steady bar in thermal conduction carries $q=400\mathrm{W}{\mathrm{m}}^{-2}$ through $A=0.01{\mathrm{m}}^2$, hence $\dot{Q}=4\mathrm{W}$; maintained for $60\mathrm{s}$, it transfers $Q=240\mathrm{J}$. This distinction makes a surface's intensity and a whole object's heating load easy to compare.

Transfer of thermal energy between regions. In a constant-property thermal Boussinesq approximation, the advective heat flux per unit area is ${\rho }_0{c}_{p}T\mathbf{u}$ and transport across a section is its normal component integrated over that section. With zero net volume transport, adding a constant reference temperature does not change that integral. Otherwise the reference and the return-flow budget must be specified. Atmospheric energy transport also includes latent heat and gravitational potential energy.

In a conventional isotropic spin-singlet superconductor, the nuclear spin-lattice relaxation rate can rise just below the transition before falling at lower temperature. The BCS coherence factors and enhanced near-gap density of states produce the coherence enhancement. Nuclear relaxation and ultrasonic attenuation have different coherence factors, so their contrasting temperature dependences test pairing structure rather than only the existence of a gap.^[ref]

A physicist.^[ref]

Helium liquefaction supplied a route to very low temperatures. The resistance-versus-temperature experiment for superconductivity and measurements of superconducting persistent current show how reaching those conditions opened new questions about electrical behavior.

A published teaching experiment used ATCC CCL-2 HeLa to make the distinction between a cell line and the condition of individual biological cells visible. Cultures were expanded in treated flasks and supplied as adherent dishes. A wash with phosphate-buffered saline preceded trypsin treatment to detach them; added fetal bovine serum stopped the digestion. The dispersed preparation was mixed with an equal volume of trypan blue and loaded into a counting chamber under an optical microscope.^[ref]

For the ethanol comparison, treatment groups and an untreated control were brought to the same final volume; the control received phosphate-buffered saline. Dye-excluding counts were compared across replicates. The paper reports representative student measurements in Figure 3A and documents the untreated control in its discussion. Its alternative temperature comparison used three replicates per condition and a $30\min$ exposure.^[ref]

The paper's disposable chamber had a different count-to-number density calibration from the hemocytometer above. Dye exclusion does not establish future division; recovery and mixing also affect counts. The constructed hemocytometer example is independent of those observations.^[ref]