Chapter Review

Thermodynamics

Gas Laws and KMT · Thermodynamics and Entropy

Kinetic Molecular Theory

KMT models gas behavior through a large number of tiny molecules in random motion undergoing perfectly elastic collisions, with no intermolecular forces except during contact.

Key Points

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Molecules are much smaller than their average separation — negligible volume assumption
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All collisions (molecule–molecule and molecule–wall) are perfectly elastic — total KE conserved
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Pressure arises from cumulative momentum transfer of molecular collisions with container walls
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Random motion means all three velocity components are equally probable: $\langle v_x^2 \rangle = \langle v_y^2 \rangle = \langle v_z^2 \rangle$
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The 1/3 factor in $P = \frac{1}{3}\rho\langle v^2 \rangle$ comes from isotropy of molecular motion

Formula

$$P = \frac{1}{3}\rho \langle v^2 \rangle$$

Temperature and Molecular KE

Absolute temperature is a direct measure of the average translational kinetic energy per molecule — heavier molecules move slower at the same temperature to maintain the same average KE.

Key Points

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Average translational KE per molecule: $\langle KE \rangle = \frac{3}{2}kT$ where $k = 1.38 \times 10^{-23}$ J/K
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At the same $T$, all gases have identical average KE regardless of molecular mass
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RMS speed: $v_{rms} = \sqrt{3kT/m} = \sqrt{3RT/M}$, so $v_{rms} \propto \sqrt{T/M}$
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Doubling $T$ doubles $\langle KE \rangle$ but only increases $v_{rms}$ by $\sqrt{2}$
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At 0 K, translational molecular motion ceases entirely

Formula

$$\langle KE \rangle = \frac{3}{2}kT$$

Ideal Gas Law and Gas Laws

The ideal gas equation of state unifies pressure, volume, temperature, and amount of gas; individual gas laws (Boyle's, Charles's) are special cases when one variable is held constant.

Key Points

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Ideal gas law: $PV = nRT$ (per mole) or $PV = NkT$ (per molecule)
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Boyle's Law ($T$ constant): $P_1V_1 = P_2V_2$ — isothermal process, PV hyperbola
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Charles's Law ($P$ constant): $V_1/T_1 = V_2/T_2$ — isobaric process
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Combined gas law: $P_1V_1/T_1 = P_2V_2/T_2$ for fixed amount of gas
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At STP (0°C, 1 atm), one mole of ideal gas occupies 22.4 L
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Temperature must always be in Kelvin for gas law calculations

Formula

$$PV = nRT$$

Internal Energy and Specific Heats

For an ideal gas, internal energy depends only on temperature (state function); the two molar specific heats $C_p$ and $C_v$ differ by exactly $R$ because heating at constant pressure also requires expansion work.

Key Points

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Monatomic ideal gas: $U = \frac{3}{2}nRT$, with $C_v = \frac{3}{2}R$ and $C_p = \frac{5}{2}R$
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Diatomic ideal gas (moderate T): $C_v = \frac{5}{2}R$, $C_p = \frac{7}{2}R$
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$C_p - C_v = R$ always holds — the extra $R$ accounts for expansion work at constant pressure
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Heat ratio $\gamma = C_p/C_v$: 5/3 for monatomic, 7/5 for diatomic
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At constant volume: all heat → internal energy ($\Delta U = C_v\Delta T$)

Formula

$$C_p - C_v = R$$

First Law of Thermodynamics

The first law is energy conservation applied to thermodynamic systems: heat added equals the increase in internal energy plus work done by the system.

Key Points

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$\Delta U = Q - W$: heat in is positive, work done by system is positive
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Internal energy ($U$) is a state function; heat ($Q$) and work ($W$) are path functions
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Isochoric ($V$ constant): $W = 0$, so $\Delta U = Q$
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Isothermal ($T$ constant): $\Delta U = 0$, so $Q = W$
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Adiabatic ($Q = 0$): $\Delta U = -W$ — work comes from internal energy
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Work at constant pressure: $W = P\Delta V$ (area under PV curve)

Formula

$$\Delta U = Q - W$$

Isothermal and Adiabatic Processes

Isothermal processes maintain constant temperature through slow heat exchange; adiabatic processes involve no heat exchange, so temperature changes as work is done.

Key Points

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Isothermal: $PV = \text{const}$ (Boyle's Law) — slow process allowing heat exchange
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Adiabatic: $PV^\gamma = \text{const}$ — rapid process or insulated system
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Adiabatic curve is steeper than isothermal on PV diagram because $\gamma > 1$
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Adiabatic expansion cools the gas; adiabatic compression heats it
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Real examples: burst tyre (adiabatic cooling), bicycle pump (adiabatic heating), cloud formation

Formula

$$PV^\gamma = \text{constant}$$

Heat Engines and Efficiency

A heat engine absorbs heat from a hot reservoir, converts part to work, and rejects the rest to a cold reservoir; the Second Law guarantees waste heat can never be zero.

Key Points

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Energy balance per cycle: $Q_H = W + Q_L$ (since $\Delta U_{\text{cycle}} = 0$)
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Thermal efficiency: $\eta = W/Q_H = 1 - Q_L/Q_H$
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Kelvin's Second Law: impossible to convert heat entirely into work from a single reservoir
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Two reservoirs at different temperatures are essential for any heat engine
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Typical efficiencies: petrol ~25-30%, diesel ~35-40%, steam turbines ~35-46%

Formula

$$\eta = 1 - \frac{Q_L}{Q_H}$$

Carnot Cycle and Maximum Efficiency

The Carnot cycle — two isothermal and two adiabatic steps — sets the theoretical upper limit on efficiency for any heat engine operating between two given temperatures.

Key Points

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Four steps: isothermal expansion at $T_H$, adiabatic expansion, isothermal compression at $T_L$, adiabatic compression
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Maximum efficiency: $\eta_{\text{Carnot}} = 1 - T_L/T_H$ (temperatures in Kelvin)
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All Carnot engines between the same two temperatures have identical efficiency regardless of working substance
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100% efficiency requires $T_L = 0$ K, which is physically unattainable
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Practical way to raise efficiency: increase $T_H$ (since $T_L$ is usually near ambient)

Formula

$$\eta_{\text{Carnot}} = 1 - \frac{T_L}{T_H}$$

Entropy and the Arrow of Time

Entropy quantifies the unavailability of energy to do work; all natural (irreversible) processes increase the total entropy of the universe, defining time's direction.

Key Points

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Entropy change (reversible): $\Delta S = \Delta Q / T$ — same heat at lower $T$ produces larger entropy change
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Reversible processes: $\Delta S_{\text{total}} = 0$; irreversible: $\Delta S_{\text{total}} > 0$
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Heat flow from $T_H$ to $T_L$: net $\Delta S = Q/T_L - Q/T_H > 0$ (always positive)
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Phase changes: melting/boiling increase entropy; freezing/condensation decrease local entropy
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Free expansion, mixing, and friction all irreversibly increase entropy
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Entropy is a state function — depends only on current state, not path

Formula

$$\Delta S = \frac{\Delta Q}{T}$$

Petrol and Diesel Engines

Real four-stroke engines approximate the Carnot ideal but with irreversible processes; diesel engines achieve higher efficiency than petrol engines due to greater compression ratios.

Key Points

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Four strokes: intake, compression, power (ignition + expansion), exhaust
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Petrol: spark ignition of fuel-air mixture; diesel: compression ignition (no spark plug)
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Higher compression ratio in diesel engines → higher operating temperature → better efficiency
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Multi-cylinder designs fire in sequence on a common crankshaft for smooth power delivery

Formulas

KMT Pressure

Gas pressure from kinetic theory — one-third density times mean square speed.

Formula

$P = \frac{1}{3}\rho \langle v^2 \rangle$

Average KE per Molecule

Average translational kinetic energy is proportional to absolute temperature.

Formula

$\langle KE \rangle = \frac{3}{2}kT$

Ideal Gas Law

Equation of state relating pressure, volume, moles, and temperature.

Formula

PV = nRT

RMS Speed

Root-mean-square molecular speed from temperature and molar mass.

Formula

$v_{rms} = \sqrt{\frac{3RT}{M}}$

First Law of Thermodynamics

Energy conservation: heat in minus work out equals change in internal energy.

Formula

$\Delta U = Q - W$

Adiabatic Condition

Pressure-volume relation when no heat is exchanged.

Formula

$PV^\gamma = \text{constant}$

Carnot Efficiency

Maximum possible efficiency using absolute reservoir temperatures.

Formula

$\eta_{\text{Carnot}} = 1 - \frac{T_L}{T_H}$

Entropy Change

Heat transferred divided by absolute temperature for a reversible process.

Formula

$\Delta S = \frac{\Delta Q}{T}$