GATE Life Sciences Thermodynamics Syllabus

What does Kirchhoff's law physically tell us about reaction enthalpy?

It tells us how the enthalpy (or internal energy) of a reaction changes with temperature, the change being governed by the difference in heat capacities of products and reactants. If $\Delta C_P > 0$, $\Delta H$ becomes more positive with rising temperature.

Define the heat (enthalpy) of reaction.

The amount of heat absorbed or released when the molar quantities of reactants indicated by the balanced equation react completely under specified conditions (usually constant $T$ and $P$). It equals $\Delta H_{reaction} = \sum \Delta H_f(\text{products}) - \sum \Delta H_f(\text{reactants})$.

Distinguish heat of reaction at constant pressure versus constant volume.

At constant pressure the heat of reaction is $q_P = \Delta H$; at constant volume it is $q_V = \Delta U$. They are related by $\Delta H = \Delta U + \Delta n_g RT$.

State Hess's law of constant heat summation.

The total enthalpy change for a reaction is the same whether it occurs in one step or several steps, provided the initial and final states are the same. It follows from $H$ being a state function.

How does Hess's law allow calculation of unknown reaction enthalpies?

By algebraically adding the enthalpy changes of a series of known reactions that combine to give the target reaction (reversing a reaction flips the sign of $\Delta H$; multiplying by a factor scales $\Delta H$). The overall $\Delta H$ equals the sum.

If $\Delta H_1$ for $A \to B$ is known, what is $\Delta H$ for $B \to A$?

$\Delta H_{B \to A} = -\Delta H_{1}$. Reversing a reaction reverses the sign of its enthalpy change (a consequence of Hess's law).

Define the standard enthalpy (heat) of formation, $\Delta H_f^{\circ}$.

The enthalpy change when one mole of a compound is formed from its constituent elements in their standard states (most stable form) at $298\ \mathrm{K}$ and $1\ \mathrm{bar}$.

What is the standard enthalpy of formation of an element in its most stable form?

It is zero by convention, e.g. $\Delta H_f^{\circ}(\ce{O2}, g) = 0$, $\Delta H_f^{\circ}(\ce{C}, graphite) = 0$.

Express the standard enthalpy of reaction in terms of enthalpies of formation.

$$\Delta H_{rxn}^{\circ} = \sum n\,\Delta H_f^{\circ}(\text{products}) - \sum n\,\Delta H_f^{\circ}(\text{reactants})$$ where $n$ are stoichiometric coefficients.

Define the standard enthalpy of combustion.

The enthalpy change when one mole of a substance is completely burned (oxidised) in excess oxygen under standard conditions. It is always negative (combustion is exothermic).

Define entropy ($S$) qualitatively.

Entropy is a state function that measures the degree of randomness or disorder of a system, or equivalently the number of accessible microstates. Higher disorder corresponds to higher entropy.

Give the thermodynamic definition of an infinitesimal entropy change for a reversible process.

$$dS = \frac{dq_{rev}}{T}$$ Entropy change equals the heat absorbed reversibly divided by the absolute temperature.

Write the Boltzmann (statistical) definition of entropy.

$$S = k_{B}\ln W$$ where $k_{B}$ is the Boltzmann constant and $W$ is the number of microstates (thermodynamic probability) corresponding to the macrostate.

State the second law of thermodynamics in terms of entropy.

For any spontaneous (irreversible) process the total entropy of the universe increases: $\Delta S_{universe} = \Delta S_{system} + \Delta S_{surroundings} > 0$. For a reversible process it stays constant.

State the third law of thermodynamics.

The entropy of a perfect, pure crystalline substance is zero at absolute zero temperature ($T = 0\ \mathrm{K}$), i.e. $\lim_{T \to 0} S = 0$. This allows absolute entropies to be determined.

Write the entropy change for an isothermal reversible expansion of an ideal gas.

$$\Delta S = nR\ln\frac{V_{2}}{V_{1}} = nR\ln\frac{P_{1}}{P_{2}}$$ Entropy increases on expansion.

Write the formula for entropy change of a phase transition at its transition temperature.

$$\Delta S = \frac{\Delta H_{trans}}{T_{trans}}$$ e.g. for fusion $\Delta S_{fus} = \Delta H_{fus}/T_{m}$ and for vaporization $\Delta S_{vap} = \Delta H_{vap}/T_{b}$.

How does entropy generally change for the physical states solid, liquid and gas?

Entropy increases in the order $S_{solid} < S_{liquid} < S_{gas}$, because disorder and the number of accessible microstates increase from solid to gas.

Predict the sign of $\Delta S$ for $\ce{2H2(g) + O2(g) -> 2H2O(l)}$.

$\Delta S < 0$ (negative). Gaseous reactants (3 mol of gas) become a liquid product with fewer moles and far less disorder.

Define Gibbs free energy ($G$) and write its defining equation.

Gibbs free energy is a state function representing the maximum non-expansion (useful) work obtainable at constant $T$ and $P$, defined as $$G = H - TS.$$

Write the equation for Gibbs free energy change at constant temperature.

$$\Delta G = \Delta H - T\Delta S$$ This is the Gibbs–Helmholtz relation used to judge spontaneity at constant $T$ and $P$.

What is the spontaneity criterion using Gibbs free energy at constant $T$ and $P$?

If $\Delta G < 0$ the process is spontaneous; if $\Delta G = 0$ the system is at equilibrium; if $\Delta G > 0$ the process is non-spontaneous (reverse is spontaneous).

Define Helmholtz free energy ($A$ or $F$).

$$A = U - TS$$ It is the maximum total work obtainable at constant temperature and volume; $\Delta A < 0$ indicates spontaneity at constant $T$ and $V$.

Make a table of $\Delta H$, $\Delta S$ signs and when a reaction is spontaneous via $\Delta G = \Delta H - T\Delta S$.

$\Delta H<0,\ \Delta S>0$: spontaneous at all $T$. $\Delta H>0,\ \Delta S<0$: non-spontaneous at all $T$. $\Delta H<0,\ \Delta S<0$: spontaneous at low $T$. $\Delta H>0,\ \Delta S>0$: spontaneous at high $T$.

At what temperature does a reaction switch between spontaneous and non-spontaneous?

At the temperature where $\Delta G = 0$, i.e. $$T = \frac{\Delta H}{\Delta S}$$ (the equilibrium/crossover temperature).

Relate standard Gibbs free energy change to the equilibrium constant.

$$\Delta G^{\circ} = -RT\ln K$$ A large positive $K$ corresponds to a large negative $\Delta G^{\circ}$.

Relate standard Gibbs free energy change to standard cell potential (electrochemistry).

$$\Delta G^{\circ} = -nFE^{\circ}_{cell}$$ where $n$ is moles of electrons transferred and $F$ is the Faraday constant. A positive $E^{\circ}_{cell}$ gives a spontaneous cell reaction.

Write the relation between $\Delta G$ under non-standard conditions and the reaction quotient $Q$.

$$\Delta G = \Delta G^{\circ} + RT\ln Q$$ At equilibrium $Q = K$ and $\Delta G = 0$, recovering $\Delta G^{\circ} = -RT\ln K$.

For an ideal gas free (Joule) expansion into vacuum, what are $w$, $q$, and $\Delta U$?

Free expansion is against zero external pressure, so $w = 0$. For an ideal gas $\Delta U = 0$ (isothermal), therefore $q = 0$. Although $q=0$, the process is irreversible and $\Delta S_{system} = nR\ln(V_2/V_1) > 0$.

Why must $\Delta H - T\Delta S$ be negative rather than just $\Delta H$ for predicting spontaneity?

Because spontaneity depends on the total entropy change of the universe. $\Delta G = \Delta H - T\Delta S$ combines the system's enthalpy (related to surroundings' entropy via $-\Delta H/T$) and the system's entropy into one criterion; $-\Delta G/T = \Delta S_{universe}$, so $\Delta G<0 \iff \Delta S_{universe}>0$.