Thermodynamics Equations

There are a lot of equations in thermodynamics. However, rather than memorising them all, think of them as to how we illustrate concepts. (pV=nRT) can be rearranged almost endlessly into various other gas laws, mole fractions and partial pressures are simply proportions, and most of the definitions of energy are summing up other definitions of energy.

Gases

Ideal gas law

The ideal gas law can be used to determine the pressure, p, volume, V, and temperature, T, of any gas. It depends only on the moles of gas, n, and not mass.

[pV = nRT]

Partial Pressures

Partial pressure is a gas-phase analogue of concentration. It is the mole fraction of a gas multiplied by the total pressure.

[p_1 = x_1 times p = frac{n_1}{n_1+n_2+…}]

Compressibility

The compressibility of a gas, Z, measures its deviation from ideality. For ideal gases, Z = 1. It comes from rearranging the ideal gas law:

[frac{pV}{nRT} = Z]

Van der Waals Equation

“Real” gases can be modelled by modifying pressure and volume to form the van der Waals equation for non-ideal gases. Two empirical parameters, a and b, are used to modify it.

[left(p + a frac{n^2}{V^2}right)left(V-nbright) = nRT]

First Law and Energy Conservation

Internal energy changes are accounted for by heat and work

[Delta U = q + w]

Internal Energy

The total energy is the sum of the kinetic and potential energy of what you’re looking at:

[U = T + V]

For a system, the internal energy is independent of the system doing any moving itself. E.g., the internal energy of hot water in a mug doesn’t include the mug doing any travelling.

Work

Work is pressure multiplied by a change in volume.

[w = – p Delta V]

Enthalpy

Enthalpy is the heat transferred (q) at constant pressure. Because volume can vary, work must be included.

[H = U + pV]

Changes in enthalpy are therefore:

[Delta H = Delta U + p Delta V]

Heat Capacity

Heat capacity is the proportionality constant between measured temperature and energy exchanged. It depends on the mass, m, of the whole system.

[q = – C_p times m times Delta T]

Second Law and Free Energy

The total entropy of the universe always increases:

[Delta S > 0]

Entropy of Reversible Processes

The heat transferred during a reversible process, q_rev, divided by temperature, T, leads to entropy.

[Delta S = frac{q_rev}{T}]

Entropy of the Surroundings

The entropy increase of the surroundings is, therefore, the enthalpy change of the system divided by temperature:

$$ Delta S = – frac{Delta H}{T}

Entropy of a Reaction (the system)

The entropy of a chemical reaction can be calculated from standard molar entropies. Remember to multiply by the stoichiometry and/or the number of moles:

[Delta S = sum n_p S_{products} – sum n_r S_{reactants}]

Free Energy

Combining the above together leads to free energy, G.

[Delta G = Delta H – T Delta S]