Problem set — thermal equilibrium

Problem set: thermal equilibrium¶

Exam-style problems. Work them with a pen before touching a computer; the last one is meant to be finished numerically. Solutions and marking rubrics live with the instructor material and are deliberately not on this site.

Problem 1 — reading the zeroth law¶

(a) State the zeroth law of thermodynamics precisely.

(b) A student argues: “If A is hotter than C, and B is hotter than C, then A and B must be at the same temperature.” Explain exactly what is wrong with this argument, and contrast it with what the zeroth law actually says.

(c) Explain, in one sentence, why the zeroth law is what makes it meaningful to calibrate a single thermometer once and then trust its reading on many different objects.

Problem 2 — the equilibrium temperature¶

A cup containing $0.20\ \mathrm{kg}$ of coffee at $90\,{}^\circ\mathrm{C}$ (specific heat $c = 4200\ \mathrm{J\,kg^{-1}\,K^{-1}}$) is poured into an empty mug of heat capacity $C_{\text{mug}} = 150\ \mathrm{J/K}$, initially at $20\,{}^\circ\mathrm{C}$. Treat the pair as isolated once the coffee is poured.

(a) Compute the coffee’s heat capacity $C_A = mc$.

(b) Compute the equilibrium temperature using the heat-capacity-weighted formula.

(c) Compare your answer with the naive unweighted average of the two starting temperatures. Which starting temperature does the true equilibrium sit closer to, and why?

Problem 3 — solving the relaxation equation¶

Two bodies with $C_A = 40\ \mathrm{J/K}$ and $C_B = 120\ \mathrm{J/K}$ are connected through a contact of conductance $\kappa = 0.50\ \mathrm{W/K}$. They start with a temperature gap of $60\ \mathrm{K}$.

(a) Derive the relaxation time $\tau$ from the pair of rate equations for $dT_A/dt$ and $dT_B/dt$, showing your algebra.

(b) Evaluate $\tau$ numerically for the values above.

(c) How long does it take for the gap to fall to $5\%$ of its initial value? Express your answer both symbolically (in terms of $\tau$) and numerically.

Problem 4 — forensic cooling¶

A body is discovered in a room held at a constant $18\,{}^\circ\mathrm{C}$. Investigators measure the body’s temperature to be $30\,{}^\circ\mathrm{C}$, and one hour later measure it again at $27\,{}^\circ\mathrm{C}$. Living human body temperature is $37\,{}^\circ\mathrm{C}$.

(a) Using $T(t) - T_{\text{room}} = (T_0 - T_{\text{room}})e^{-t/\tau}$, use the two measurements to solve for $\tau$.

(b) Using $\tau$ from (a) and an assumed starting temperature of $37\,{}^\circ\mathrm{C}$ at the moment of death, estimate how long before the first measurement death occurred.

(c) Name two assumptions of the model that are most likely to be violated in a real forensic setting (consider: is $\kappa$ really constant? is the room temperature really constant? is $37\,{}^\circ\mathrm{C}$ a safe assumption for every individual?), and say which direction each violation would bias the estimated time of death.

Problem 5 — where the model breaks¶

The laboratory’s discrete exchange model assumes both bodies sit in the classical, high-temperature equipartition regime and exchange quanta of equal size.

(a) Using the model specification’s stated failure modes, explain what goes wrong as both bodies’ temperatures approach absolute zero.

(b) Explain why a body with only a handful of oscillators (say $n_A = 2$) makes “temperature” a shaky concept even before quantum statistics enter the picture. Connect your answer to the rounding step in from_temperatures described in the page’s Advanced section.

(c) Two different real materials in direct contact do not literally exchange quanta of exactly equal size. Give a physical argument for why this does not stop the macroscopic conclusions of this module (the equilibrium temperature formula, the exponential relaxation) from holding, even though it does invalidate the specific microscopic mechanism modelled in the laboratory.

Problem 6 — computational¶

Using the module’s laboratory notebook and the thermolab.equilibrium module:

(a) Build a state with from_temperatures, choosing your own $n_A \ne n_B$ and starting temperatures. Run many independent long simulations and verify that the mean final temperature agrees with equilibrium_temperature, quoting an uncertainty on your estimate.

(b) At several checkpoints (roughly $0.5\tau$, $1\tau$, $2\tau$ and $4\tau$), measure the mean temperature gap across independent seeds and compare it with predicted_relaxation. Does the agreement hold at every checkpoint, or only near equilibrium?

(c) Repeat the exercise at two different system sizes (say ten times apart in oscillator count) and measure the run-to-run relative spread of the gap one relaxation time in. Verify that the spread shrinks as the system grows.

(d) State one thing your numerical result does not prove about real thermal contact.