Thermodynamics Equation Sheet

Ideal Gas Law

• Equation of state for a dilute gas without intermolecular forces:

  • $pV = Nk_BT$, where:

    • $p$ is pressure.

    • $V$ is volume.

    • $N$ is the number of molecules.

    • $k_B$ is the Boltzmann constant.

      • $k_B=\frac{R}{N_A}$

      • $R$ is the Universal Gas Constant ($8.315 \text{ J mol}^{-1}\text{ K}^{-1}$)

    • $T$ is temperature.

  • Alternative forms:

    • $p = nk_BT$, where $n = N/V$ is the number density of molecules.

    • $pV = n_mRT$, where $n_m$ is the number of moles of molecules and $R$ is the Universal Gas Constant.

Heat Capacity

• Proportionality between heat added to an object and the change in its temperature.

• $C_x = \frac{dQ}{dT}|_x$, where:

  • $dQ$ is an infinitesimal addition of heat.

  • $dT$ is an infinitesimal change in temperature.

  • $x$ is the quantity held constant (either $p$ or $V$).

• Specific heat capacity:

  • $c_x = \frac{1}{M} \frac{dQ}{dT}|_x$, where $M$ is the mass of the object.

Definition of Temperature

• A rigorous definition of temperature:

  • $\frac{1}{k_BT} = \frac{d \ln \Omega}{dE}$ , where $\Omega$ is the multiplicity (the number of microstates corresponding to the observed macrostate of the system).

Boltzmann Distribution

• Probability distribution for the energy of a molecule coupled to a large reservoir (e.g., a gas at temperature $T$):

  • $P(\epsilon) = Ae^{-\epsilon/(k_BT)}$, where:

    • $A$ is a constant.

    • $\epsilon$ is the energy of the molecule.

Maxwell-Boltzmann Distribution

• Probability distribution of molecular speed in a gas at temperature $T$:

  • $P(v) = 4\pi \left( \frac{m}{2\pi k_BT} \right)^{3/2} v^2 \exp \left( -\frac{mv^2}{2k_BT} \right)$, where:

    • $m$ is the mass of a single molecule.

    • $v$ is the speed.

Mean Kinetic Energy

• The average kinetic energy of a molecule in a gas at temperature $T$:

  • $\langle E_{KE} \rangle = \frac{3}{2} k_BT$

Mean Free Path

• The average distance a molecule in a gas travels between collisions:

  • $\lambda = (\sqrt{2} n \sigma)^{-1}$

  • $\sigma$ is the molecular cross-sectional area.

Conduction and Diffusion

• Heat flux through a gas in the x-direction:

  • $J_x = -\kappa \frac{\partial T}{\partial x}$

    • $\kappa$ is the thermal conductivity.

• Diffusion equation in the x-direction:

  • $\frac{\partial T}{\partial t} = D \frac{\partial^2 T}{\partial x^2}$,

    • $D$ is the diffusivity.

• Chemical diffusion:

  • $\Phi_x = -D \frac{\partial n^*}{\partial x}$

    • $n^*$ is the number density of the diffusing molecules.

  • $\frac{\partial n^*}{\partial t} = D \frac{\partial^2 n^*}{\partial x^2}$

Function of State

• A variable describing a system in thermodynamic equilibrium that is path-independent.

• For a change from state a to state b:

  • $f = \int_{x_a}^{x_b} df = f(x_b) - f(x_a)$, where $x$ is a list of system parameters (e.g., pressure, temperature).

  • $df$ must be an exact differential.

First Law of Thermodynamics

• Energy is conserved; heat and work are both forms of energy:

  • $dU = dQ + dW$, where:

    • $dU$ is an infinitesimal change in internal energy.

    • $dQ$ is an infinitesimal quantity of heat added to the system.

    • $dW$ is an infinitesimal quantity of work done by the system.

Work

• Work is force times distance.

  • In a thermodynamic system:

    • $dW = -p dV$

    • $dV$ is an infinitesimal change in volume.

Reversible Isothermal Expansion of an Ideal Gas

• For an ideal gas, $U = U(T)$, so if $dT = 0$, then $dU = 0$ and $dQ = -dW$.

• Using the ideal gas law:

  • $\Delta Q = RT \ln \frac{V_2}{V_1}$ per mole of gas.

Adiabatic Expansion of an Ideal Gas

• Adiabatic means reversible and adiathermal ($dQ = 0$).

• Using this and $dU = C_V dT$ for an ideal gas, and the ideal gas law:

  • $pV^\gamma = \text{const}$,

    • $\gamma$ is the adiabatic index, which is $\gamma=\frac{C_p}{C_v}$.

Efficiency of an Engine Cycle

• Engines convert heat into work.

• Efficiency is given by:

  • $\eta = \frac{W}{Q_h} = \frac{Q_h - Q_c}{Q_h}$, where:

    • $Q_h$ is the heat input.

    • $Q_c$ is the waste heat output.

Efficiency of a Carnot Engine

• The Carnot engine is the most efficient engine possible between two thermal reservoirs.

• It is composed of two isothermal paths and two adiabatic paths.

• Its efficiency is:

  • $\eta_{Carnot} = 1 - \frac{Tc}{T_h}$, where:

    • $T_h$ is the temperature of the hot reservoir.

    • $T_c$ is the temperature of the cold reservoir.

Second Law of Thermodynamics

• No process is possible with the sole result of complete conversion of heat into work.

• Equivalently, no process is possible with the sole result of transferring heat from a colder to a hotter body.

Entropy

• A function of state defined by the exact differential:

  • $dS = \frac{dQ_{rev}}{T}$

    • $dQ_{rev}$ is the amount of heat added to the system reversibly.

• For any process:

  • $dS \geq 0$, where the equality holds if the process is reversible.

• The definition of entropy from kinetic theory is:

  • $S = k_B \ln \Omega$, which is consistent with the definition of temperature from kinetic theory.

Latent Heat

• The energy associated with a phase change is called the latent heat.

• It is given by:

  • $L = T \Delta S$, where:

    • $T$ is the temperature of the phase change.

    • $\Delta S$ is the entropy difference between the two phases.

Clausius-Clapeyron Equation

• This equation describes the slope of phase boundaries on a phase diagram:

  • $\frac{dp}{dT} = \frac{L}{T\Delta V}$, where $\Delta V$ is the change in volume (or the change in specific volume) between the two phases.