Physics: Heat and Thermodynamics

Physics: Heat and Thermodynamics

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Heat is energy in transit; temperature is the scalar state variable that determines the direction of spontaneous heat flow. The Zeroth Law defines temperature through thermal equilibrium, the First Law (ΔU = Q − W) conserves energy in any process, and the Second Law fixes the direction via entropy (ΔS ≥ 0 for an isolated system). Work done by an ideal gas is W = PΔV in an isobaric process and W = nRT ln(V₂/V₁) isothermally; in an adiabatic process PV^γ = constant with γ = Cp/Cv. Carnot efficiency η = 1 − T_cold/T_hot is the ceiling no real engine can exceed, and latent heat (Q = mL) must be added whenever a phase change occurs during heating. Always use Kelvin in PV = nRT and in Carnot-efficiency calculations — plugging Celsius here is the most common point loss.

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🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Defining the Key Quantities

Temperature (T, in K) is a macroscopic property that attains the same value for any two systems in mutual thermal equilibrium — the content of the Zeroth Law. Heat (Q, in J) is energy crossing the boundary of a system because of a temperature difference; internal energy U is the total microscopic kinetic plus potential energy stored inside. For an ideal gas U depends only on T, and ΔU = nCvΔT.

Heat Transfer Mechanisms

Energy moves three ways: conduction obeys Fourier’s law (Q/t = −kA dT/dx), where k is the thermal conductivity; convection transports heat through bulk fluid motion; radiation follows the Stefan–Boltzmann law (P = εσAT⁴), where σ = 5.67 × 10⁻⁸ W m⁻² K⁻⁴. Sensible heating uses Q = mcΔT, but a phase change requires Q = mL (latent heat) with no temperature change — students who omit this in a melting/boiling calculation lose full marks.

First Law and the Four Processes

ΔU = Q − W, where W is work done by the system.

Process	Constant	Work by gas	ΔU
Isobaric	P	PΔV	nCpΔT
Isochoric	V	0	nCvΔT
Isothermal	T	nRT ln(V₂/V₁)	0
Adiabatic	Q = 0	(P₁V₁ − P₂V₂)/(γ − 1)	nCvΔT

The Second Law states that entropy of an isolated system never decreases: ΔS = Q/T for reversible paths. This rules out perpetual-motion machines and explains why Carnot efficiency η = 1 − T_cold/T_hot (T in Kelvin) is the upper bound for any engine operating between two reservoirs.

Typical HAT-UG Question Patterns

• Numerical: compute W, Q, ΔU for a gas taken through a cyclic P–V diagram.
• Conceptual: identify which law forbids a 100% efficient engine.
• Conversion: convert Celsius to Kelvin before applying PV = nRT.

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Edge Cases and Mechanism Depth

The adiabatic lapse of PV^γ arises from combining the First Law (dU = δQ − δW = −PdV) with U = nCvT and PV = nRT, eliminating temperature to leave the differential form γ dV/V + dP/P = 0. In free expansion of an ideal gas into a vacuum, W = 0, Q = 0, so ΔU = 0 — yet entropy increases because the process is irreversible; the Second Law is violated only if you mistakenly equate “no heat exchange” with “ΔS = 0”.

Specific heat is not constant: cp and cv vary with temperature (especially near phase transitions), and Cp − Cv = R holds only for ideal gases. For solids at low T the Debye T³ law replaces classical Dulong–Petit predictions.

Connections to Adjacent Topics

Heat and Thermodynamics feeds directly into Kinetic Theory (mean kinetic energy per molecule = 3kT/2) and into Waves (sound speed a = √(γRT/M) for an ideal gas). The sign convention of W is also why the work-energy theorem in mechanics carries a sign: work done on a gas is −PΔV.

Common Mistakes in HAT-UG

• Using 273 instead of 273.15 when precision is asked.
• Reading η = 1 − T_cold/T_hot as a percentage without converting the fraction.
• Applying isothermal formulas (PV = const) to an adiabatic process.
• Forgetting that entropy change of a reservoir is −Q_res/T, so total ΔS_universe ≥ 0 even when ΔS_system < 0.

Practice Prompts

1. Two moles of an ideal monatomic gas expand isothermally at 400 K from 2 L to 8 L. Find Q, W, and ΔU, and verify the First Law.
2. A Carnot engine operates between 500 K and 300 K with 1 kJ of heat drawn from the hot reservoir. Compute the work output and the entropy change of each reservoir; show ΔS_universe ≥ 0.

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