The average translational kinetic energy of gas molecules is directly proportional to absolute temperature:

$\langle K \rangle = \frac{3}{2}k_B T = \frac{3}{2} \frac{R}{N_A} T$

Where:

• $k_B = 1.38 \times 10^{-23}$ J/K (Boltzmann constant)
• R = 8.314 J/(mol - K) (universal gas constant)
• T = absolute temperature in Kelvin

This means temperature is a measure of the average kinetic energy of particles.

The distribution of molecular speeds in a gas follows the Maxwell-Boltzmann distribution:

$f(v) = 4\pi \left(\frac{m}{2\pi k_B T}\right)^{3/2} v^2 e^{-mv^2/(2k_B T)}$

Key features:

• Most probable speed: $v_p = \sqrt{\frac{2k_BT}{m}}$
• Average speed: $\bar{v} = \sqrt{\frac{8k_BT}{\pi m}}$
• RMS speed: $v_{rms} = \sqrt{\frac{3k_BT}{m}}$
• Distribution is broader at higher temperatures
• Lighter molecules have higher speeds than heavier ones at same T

The root-mean-square speed of gas molecules:

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

Where:

• m = mass of individual molecule
• M = molar mass of gas

The rms speed increases with temperature and decreases with molecular mass.

The ideal gas model is based on the following assumptions:

• Gas consists of large number of tiny particles (atoms or molecules)
• Particles are in constant random motion
• Particles are point masses with negligible volume
• Collisions between particles and walls are perfectly elastic
• No intermolecular forces (except during collisions)
• Motion is governed by Newton's laws

The ideal gas law relates pressure, volume, amount of gas, and temperature:

$PV = nRT$

Where:

• P = absolute pressure (Pa)
• V = volume (m3)
• n = number of moles
• R = 8.314 J/(mol - K)
• T = absolute temperature (K)

Alternative form using number of molecules: $PV = Nk_B T$

Where N = number of molecules and $k_B$ = Boltzmann constant.

For a fixed amount of gas (constant n):

$\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}$

Special cases:

• Boyle's Law (constant T): $P_1 V_1 = P_2 V_2$ - Pressure inversely proportional to volume
• Charles's Law (constant P): $\frac{V_1}{T_1} = \frac{V_2}{T_2}$ - Volume directly proportional to temperature
• Gay-Lussac's Law (constant V): $\frac{P_1}{T_1} = \frac{P_2}{T_2}$ - Pressure directly proportional to temperature

One mole contains exactly Avogadro's number of particles: $N_A = 6.022 \times 10^{23}$ particles/mol

Number of molecules N and moles n are related by: $N = n N_A$

Molar mass M relates to molecular mass m: $M = m N_A$

Conduction is heat transfer through direct molecular contact within a material.

Rate of heat transfer by conduction:

$\frac{Q}{t} = kA\frac{\Delta T}{L}$

Where:

• k = thermal conductivity coefficient (W/m - K)
• A = cross-sectional area
• $\Delta T$ = temperature difference
• L = length/thickness of material

Materials with high k (metals) conduct heat well; insulators have low k.

Convection is heat transfer by bulk motion of fluid.

• Natural convection: Fluid motion caused by density differences due to temperature (hot fluid rises)
• Forced convection: Fluid motion caused by external forces (fans, pumps)

Rate depends on:

• Temperature difference
• Surface area
• Fluid properties (density, viscosity, specific heat)
• Flow velocity

Radiation is heat transfer by electromagnetic waves. No medium required.

Stefan-Boltzmann law:

$P = \sigma \varepsilon A T^4$

Where:

• $\sigma = 5.67 \times 10^{-8}$ W/(m2 - K4) (Stefan-Boltzmann constant)
• $\varepsilon$ = emissivity (0 ≤ $\varepsilon$ ≤ 1)
• A = surface area
• T = absolute temperature

Net radiation between two objects: $P_{net} = \sigma \varepsilon A (T^4 - T_{surroundings}^4)$

Thermal equilibrium is reached when:

1. All objects in contact reach the same temperature
2. No net heat flow occurs between objects

Zeroth Law of Thermodynamics: If object A is in thermal equilibrium with B, and B with C, then A is in thermal equilibrium with C. This allows us to define temperature consistently.

When equilibrium is reached: $T_A = T_B = T_C = ...$ $\frac{dQ}{dt} = 0$ (no net heat flow)

The First Law of Thermodynamics (conservation of energy for thermodynamic systems):

$\Delta U = Q - W$

Where:

• $\Delta U$ = change in internal energy
• Q = heat added to system (+) or removed (-)
• W = work done BY the system (+) or ON the system (-)

Sign convention summary:

• Q > 0: Heat added to system
• W > 0: Work done by system (expansion)
• $\Delta U$ > 0: Internal energy increases

For an ideal gas, internal energy depends only on temperature: $U = \frac{f}{2}nRT$ (where f = degrees of freedom)

Work done by gas during expansion/compression:

$W = \int_{V_1}^{V_2} P \, dV$

For constant pressure processes: $W = P \Delta V = P(V_2 - V_1)$

Work sign:

• Expansion ($V_2 > V_1$): W > 0 (gas does work)
• Compression ($V_2 < V_1$): W < 0 (work done on gas)

The work done by a gas equals the area under the P-V curve:

$W = \int P \, dV = \text{Area under P-V curve}$

• Area enclosed by a cycle = net work for cycle
• Clockwise cycle: net work done by system (+)
• Counterclockwise cycle: net work done on system (-)

This is why PV diagrams are so useful for calculating work.

Volume remains constant during the process.

Characteristics:

• $V$ = constant
• Work: $W = 0$ (no volume change, no work)
• From ideal gas law: $P/T$ = constant
• First law: $\Delta U = Q$

On PV diagram: vertical line

Temperature remains constant during the process.

Characteristics:

• $T$ = constant
• For ideal gas: $\Delta U = 0$ (U depends only on T)
• First law: $Q = W$ (heat added equals work done)
• PV relationship: $PV$ = constant
• Work: $W = nRT\ln(V_2/V_1) = nRT\ln(P_1/P_2)$

On PV diagram: $PV = \text{constant}$ hyperbola

Pressure remains constant during the process.

Characteristics:

• $P$ = constant
• Work: $W = P\Delta V = P(V_2 - V_1)$
• From ideal gas law: $V/T$ = constant
• First law: $\Delta U = Q - P\Delta V$

On PV diagram: horizontal line

No heat transfer to/from the system ($Q = 0$).

Characteristics:

• $Q = 0$ (insulated system)
• First law: $\Delta U = -W$ (work done at expense of internal energy)
• For ideal gas: $TV^{\gamma-1}$ = constant, $PV^{\gamma}$ = constant
• Work: $W = \frac{P_2V_2 - P_1V_1}{1-\gamma} = nC_v\Delta T$

Where $\gamma = C_p/C_v$ (ratio of specific heats)

• Monatomic gas: $\gamma = 5/3 ≈ 1.67$
• Diatomic gas: $\gamma = 7/5 = 1.4$

A cyclic process returns the system to its initial state.

Characteristics:

• $\Delta U = 0$ (returns to initial state)
• Net work: $W_{net} = Q_{net} = Q_{in} - Q_{out}$
• $\Delta U = Q - W = 0 \Rightarrow Q = W$

On PV diagram: closed loop

• Area inside loop = net work for cycle
• Clockwise: positive work (engine)
• Counterclockwise: negative work (refrigerator/heat pump)

Specific Heat Capacity Formula Q=mcDeltaT:

The amount of heat required to change temperature:

$Q = mc\Delta T$

Where:

• Q = heat added (+) or removed (-)
• m = mass
• c = specific heat capacity (J/kg - K)
• $\Delta T = T_{final} - T_{initial}$

Molar specific heat capacity: $Q = nC\Delta T$

Where C = molar heat capacity (J/mol - K)

Different values for different processes:

• $C_v$: specific heat at constant volume
• $C_p$: specific heat at constant pressure
• $C_p > C_v$ (work done during expansion at constant P)

determines how well a material conducts heat.

Typical values (W/m - K):

• Metals: high (copper ≈ 400, aluminum ≈ 237)
• Liquids: moderate (water ≈ 0.6)
• Gases: low (air ≈ 0.024)
• Insulators: very low (fiberglass ≈ 0.04)

High k: good conductors (heat flows quickly) Low k: good insulators (heat flows slowly)

Calorimetry measures heat transfer using conservation of energy.

Principle: In an isolated system, total energy is conserved.

$Q_{gained} + Q_{lost} = 0$ $m_1c_1\Delta T_1 + m_2c_2\Delta T_2 + ... = 0$

Applications:

• Measuring specific heat capacities
• Determining heat of reactions
• Finding mixing temperatures

Example (mixing hot and cold water): $m_{hot}c(T_{final} - T_{hot}) + m_{cold}c(T_{final} - T_{cold}) = 0$

During phase changes, temperature remains constant while heat is added or removed.

$Q = mL$

Where:

• m = mass of substance changing phase
• L = latent heat (J/kg)

Types of latent heat:

• Heat of fusion ($L_f$): melting/freezing (solid <-> liquid)
• Heat of vaporization ($L_v$): boiling/condensation (liquid <-> gas)

Typical values (water):

• $L_f = 3.33 \times 10^5$ J/kg
• $L_v = 2.26 \times 10^6$ J/kg

Heating/cooling curve: temperature stays constant during phase change

Entropy (S) is a measure of disorder or randomness of a system.

For a reversible process: $\Delta S = \frac{Q_{rev}}{T}$

For infinitesimal changes: $dS = \frac{dQ_{rev}}{T}$

Units: J/K

Entropy change for system + surroundings: $\Delta S_{universe} = \Delta S_{system} + \Delta S_{surroundings}$

Reversible process:

• System and surroundings can be returned to initial states
• Quasi-equilibrium throughout (infinitely slow)
• $\Delta S_{universe} = 0$

Irreversible process:

• Cannot return to original state without external changes
• Occurs naturally (all real processes)
• $\Delta S_{universe} > 0$

Examples of irreversible processes:

• Free expansion of gas
• Heat flow from hot to cold
• Mixing of different substances
• Friction

Heat spontaneously flows from hot to cold, never reverse.

Mathematical expression: $\Delta S_{hot} = \frac{-Q}{T_{hot}}$ (heat leaves hot) $\Delta S_{cold} = \frac{Q}{T_{cold}}$ (heat enters cold) $\Delta S_{universe} = Q\left(\frac{1}{T_{cold}} - \frac{1}{T_{hot}}\right) > 0$

Since $T_{cold} < T_{hot}$, $\Delta S_{universe} > 0$

This is a direct consequence of the second law.

Heat engine converts thermal energy to mechanical work.

$Q_H \rightarrow \text{Engine} \rightarrow W + Q_C$

Efficiency: $e = \frac{W}{Q_H} = \frac{Q_H - Q_C}{Q_H} = 1 - \frac{Q_C}{Q_H}$

Where:

• $Q_H$ = heat absorbed from hot reservoir
• $Q_C$ = heat rejected to cold reservoir
• W = net work output

Carnot efficiency (maximum possible, reversible engine): $e_{Carnot} = 1 - \frac{T_C}{T_H}$

Where temperatures are in Kelvin.

Second law implications:

• No engine can be 100% efficient ($e < 1$)
• Carnot efficiency is theoretical maximum
• $e_{actual} < e_{Carnot}$ for all real engines