Heat, Temperature and Thermodynamics

Heat, Temperature and Thermodynamics

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

• Temperature measures the average kinetic energy of molecules; it is a state property, not a quantity of energy.
• Heat (Q) is thermal energy in transit between bodies at different temperatures — once absorbed it becomes internal energy.
• The single must-remember equations are Q = mcΔT (sensible heat), Q = mL (latent heat), and the First Law ΔU = Q − W with W as work done by the system.
• The Carnot efficiency η = 1 − T_c/T_h requires absolute (Kelvin) temperatures; using °C gives a wrong answer.
• Watch the trap: during a phase change temperature is constant, so Q = mcΔT does not apply — switch to Q = mL.

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🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Defining Heat vs Temperature vs Internal Energy

Temperature (T) is the degree of hotness that tells us whether two systems are in thermal equilibrium (Zeroth Law of Thermodynamics). Heat (Q) is energy crossing the boundary of a system purely because of a temperature gradient. Internal energy (U) is the total kinetic + potential energy of all molecules inside the system; it is a state function, so dU depends only on the initial and final states, not the path.

Calorimetry: Q = mcΔT and Q = mL

The energy needed to raise 1 kg of a substance by 1 K is its specific heat capacity c (J kg⁻¹ K⁻¹), giving Q = mcΔT (sensible heat). During a phase change (melting, boiling, sublimation) the temperature stays pinned at the transition value; the heat absorbed or released is Q = mL, where L is the specific latent heat (J kg⁻¹). ECAT MCQs frequently combine the two: warming ice from −10 °C to 0 °C uses Q = mcΔT, melting it uses Q = mL, then warming the water again uses Q = mcΔT with water’s c.

Thermal Expansion

Solids expand when heated. Linear expansion ΔL = αL₀ΔT, areal ΔA = βA₀ΔT, and volumetric ΔV = γV₀ΔT, where α, β, γ are coefficients (K⁻¹) with the approximate relation β ≈ 2α and γ ≈ 3α for isotropic solids. A bimetallic strip and a mercury thermometer both exploit these coefficients.

First Law of Thermodynamics

ΔU = Q − W where W is work done by the system. For an ideal gas, the boundary work during an isobaric process is W = PΔV; for an isothermal expansion at temperature T, W = nRT ln(V_f/V_i); for an adiabatic process, PV^γ = constant and W = (P_iV_i − P_fV_f)/(γ − 1). The Mayer relation C_p − C_v = R and γ = C_p / C_v (≈ 1.67 for monatomic, 1.4 for diatomic gases) are tested almost every year.

Second Law and Carnot Efficiency

Heat flows spontaneously from hot to cold. No engine operating between two reservoirs can exceed the Carnot efficiency η = 1 − T_c / T_h, with temperatures in kelvin. ECAT numerics often ask for η as a percentage, so convert at the end: multiply by 100.

Typical ECAT Patterns

• Numerical calorimetry combining solid + liquid + phase change.
• Sign-convention MCQs on ΔU = Q − W.
• Computing W for isothermal vs adiabatic expansion of an ideal gas.
• Carnot-efficiency calculation requiring Kelvin conversion.

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Edge Cases and Subtleties

The Zeroth Law is what justifies the existence of a thermometer: if A is in equilibrium with C, and B is in equilibrium with C, then A and B are in equilibrium with each other — allowing us to define a universal temperature scale. The Celsius-to-Kelvin shift T(K) = T(°C) + 273.15 is the most common source of lost marks in ECAT Carnot problems; a 27 °C reservoir (300 K) vs a 327 °C reservoir (600 K) gives η = 1 − 300/600 = 50 %, not 1 − 27/327.

For real substances, c itself varies with temperature, so the integrated form Q = ∫ mc(T) dT is more accurate; but ECAT restricts you to constant-c values. Latent heat splits into latent heat of fusion (L_f) at the solid–liquid boundary and latent heat of vaporisation (L_v) at the liquid–gas boundary, with L_v ≫ L_f for most substances because vaporisation breaks all intermolecular bonds, not just some.

Connections to Adjacent Topics

The First Law is conservation of energy restated for thermodynamic systems; it links directly to work-energy theorem in mechanics. Entropy (S), defined as dS = dQ_rev/T, quantifies the Second Law — ΔS_universe ≥ 0 — and connects heat to statistical mechanics via S = k_B ln Ω. The four thermodynamic processes (isothermal, adiabatic, isobaric, isochoric) are best visualised on a P–V diagram, where the area under the curve equals work done.

Common Mistakes

• Using Q = mcΔT during boiling — temperature is constant, so ΔT = 0 and the formula gives zero, which is wrong.
• Writing ΔU = Q + W after defining W as work done on the system (pick one convention and stick to it).
• Computing η = (T_h − T_c)/T_h in Celsius units instead of kelvin.
• Treating c and C (specific vs molar heat capacity) as interchangeable; C = Mc.
• Assuming all gases have γ = 1.4 — noble gases are monatomic with γ ≈ 5/3.

Practice Prompts

1. Numerical: 200 g of ice at −10 °C is converted to steam at 100 °C. Using c_ice = 2100 J kg⁻¹ K⁻¹, c_water = 4200 J kg⁻¹ K⁻¹, L_f = 3.34 × 10⁵ J kg⁻¹, L_v = 2.26 × 10⁶ J kg⁻¹, find the total heat absorbed.
2. Conceptual: A Carnot refrigerator operates between 270 K and 300 K. State, with reasoning, whether its coefficient of performance equals (T_c / (T_h − T_c)) — derive it from η = 1 − T_c/T_h.

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