10.2 Heat, Temperature & Gas Laws

10.2 Heat, Temperature & Gas Laws

Quick Answer: Temperature is a measure of hotness tied to average molecular kinetic energy; heat is energy in transit because of a temperature difference. Use Q = mcΔT for sensible heating/cooling and Q = mL for phase changes. For gases at FSc level, apply PV = nRT with T in kelvin, and specialize to Boyle (T fixed), Charles (P fixed), and Gay-Lussac (V fixed)—all of which appear in atmospheric, tire, and engine-intake reasoning.

Aeronautical systems constantly exchange heat and manage gases under changing pressure and temperature. Turbine engines ingest air whose density depends on T and p; landing-gear struts and tires hold compressed gas; the International Standard Atmosphere (ISA) models how temperature falls with altitude. This section locks the thermal vocabulary and gas laws you need for exam calculations.

Temperature vs Heat

Temperature indicates how hot or cold a body is. On the microscopic picture taught at FSc level, absolute temperature is proportional to the average translational kinetic energy of molecules. Scales:

Heat is energy transferred between systems because of a temperature difference—not “something a body contains” in modern language (internal energy U is the stored microscopic energy). SI unit of heat and energy: joule (J). Older texts use calorie (1 cal ≈ 4.186 J).

Heat flows spontaneously from higher to lower temperature until thermal equilibrium—setting up the zeroth law discussed in the next section.

Specific Heat Capacity

When a body changes temperature without changing phase:

$Q = mc\Delta T$

• m — mass
• c — specific heat capacity (J kg⁻¹ °C⁻¹ or J kg⁻¹ K⁻¹; the size of a degree is the same)
• ΔT — temperature change

Water has a high c (≈ 4180 J kg⁻¹ °C⁻¹), so it stores a lot of thermal energy per degree—useful for cooling jackets and for understanding why large water masses change temperature slowly. Metals used in engines and airframes have much lower c and heat (or cool) faster for the same mass and heat input.

Worked example — specific heat. How much heat raises 2.0 kg of aluminum (c ≈ 900 J kg⁻¹ °C⁻¹) from 20 °C to 120 °C?
Q = mcΔT = 2.0 × 900 × (100) = 1.8 × 10⁵ J.

Latent Heat and Phase Change

During melting or boiling at constant pressure, temperature stays constant while heat is absorbed or released:

$Q = mL$

L_f = latent heat of fusion; L_v = latent heat of vaporization. Water: L_f ≈ 3.34 × 10⁵ J/kg, L_v ≈ 2.26 × 10⁶ J/kg. Evaporative cooling (sweat, fuel vaporization in some contexts, fog formation) is latent-heat physics.

Worked example — latent heat. Energy to melt 0.50 kg of ice at 0 °C: Q = 0.50 × 3.34 × 10⁵ = 1.67 × 10⁵ J—with no temperature rise until melting finishes.

Thermal Expansion (Brief Aero Link)

Linear expansion: ΔL = αL₀ΔT. Area and volume expansions use 2α and ≈3α for isotropic solids. Riveted joints, control cables, and fuel volumes all change with temperature; designers leave expansion gaps and specify temperature ranges. Gases expand much more dramatically—captured by Charles’s law below.

Ideal Gas Model and Absolute Temperature

An ideal gas obeys:

$PV = nRT$

• P — absolute pressure (Pa)
• V — volume (m³)
• n — amount of substance (mol)
• R — universal gas constant ≈ 8.314 J mol⁻¹ K⁻¹
• T — absolute temperature (K)

Equivalently pV = NkT with Boltzmann’s constant k. Real air at ordinary aircraft temperatures and pressures is close enough to ideal for FSc problems unless the question states otherwise.

Critical rule: never insert Celsius into PV = nRT. Convert: T(K) = t(°C) + 273 (more precisely +273.15; exams usually accept +273).

Boyle’s, Charles’s, and Gay-Lussac’s Laws

Special cases of the ideal gas law:

Worked example — Boyle. Gas in a cylinder at 2.0 × 10⁵ Pa occupies 0.030 m³ at constant temperature. Compressed to 0.010 m³, new pressure P₂ = P₁V₁/V₂ = (2.0 × 10⁵)(0.030)/0.010 = 6.0 × 10⁵ Pa.

Worked example — Charles. Air at constant pressure occupies 0.020 m³ at 27 °C (300 K). Heated to 127 °C (400 K): V₂ = V₁(T₂/T₁) = 0.020 × (400/300) = 0.0267 m³.

Worked example — Gay-Lussac (tire-style). A closed rigid volume holds gas at 250 kPa absolute and 20 °C (293 K). After taxi, temperature reaches 50 °C (323 K). P₂ = P₁(T₂/T₁) = 250 × (323/293) ≈ 275 kPa. Pressure rose though volume was fixed—exactly why tire and strut pressures are checked cold with temperature corrections in mind.

Worked example — combined / atmosphere flavor. A mass of air at 101 kPa and 15 °C (288 K) is taken to a condition at 70 kPa and −5 °C (268 K). Volume ratio: V₂/V₁ = (P₁/P₂)(T₂/T₁) = (101/70)(268/288) ≈ 1.34. Same air mass occupies more volume aloft when pressure falls faster than temperature in this scenario—density ρ = m/V therefore drops, cutting lift for a given true airspeed.

Density, ISA Thinking, and Engines

From the ideal gas law, ρ = PM/(RT) for molar mass M—so density rises with pressure and falls with temperature. That is the heart of density altitude: hot + high → lower ρ → less lift and thrust for the same indicated conditions. Intake mass flow ṁ ≈ ρAv likewise falls when density falls, affecting engine power.

Heat Transfer Modes (Exam Awareness)

• Conduction — through solids (engine mounts, skin)
• Convection — bulk fluid motion (cooling airflow over fins and radiators)
• Radiation — electromagnetic emission (skin in sunlight, hot turbine parts)

Questions may ask which mode dominates in a scenario; link the mode to the physical path of energy, not memorized slogans alone.

Common Mistakes

• Using °C inside PV = nRT
• Mixing gauge pressure with absolute pressure in gas laws
• Applying Q = mcΔT across a phase change (use mL instead while T is constant)
• Forgetting that Boyle requires constant temperature—fast compressions are closer to adiabatic processes (advanced topic; see thermodynamics laws section for heat engines)

If you can move fluently among Q = mcΔT, Q = mL, and P₁V₁/T₁ = P₂V₂/T₂ with kelvin temperatures, you own this blueprint slice.