GATE Chemical Engineering Process Calculations and Thermodynamics Syllabus

Write the entropy change of an ideal gas between two states in terms of $T$ and $P$.

$$\Delta S = C_P \ln\frac{T_2}{T_1} - R \ln\frac{P_2}{P_1}$$

State the entropy balance for a steady-flow open system.

$$\Delta(S\dot m) = \sum \frac{\dot Q_j}{T_j} + \dot S_{gen}, \qquad \dot S_{gen} \geq 0$$

Give the efficiency of a Carnot engine operating between $T_H$ and $T_C$.

$$\eta_{Carnot} = 1 - \frac{T_C}{T_H}$$ (temperatures in absolute units).

Write the four fundamental property (Gibbs) relations for a closed system of constant composition.

$$dU = T\,dS - P\,dV$$ $$dH = T\,dS + V\,dP$$ $$dA = -S\,dT - P\,dV$$ $$dG = -S\,dT + V\,dP$$

State the four Maxwell relations.

$$\left(\frac{\partial T}{\partial V}\right)_S = -\left(\frac{\partial P}{\partial S}\right)_V,\quad \left(\frac{\partial T}{\partial P}\right)_S = \left(\frac{\partial V}{\partial S}\right)_P$$ $$\left(\frac{\partial S}{\partial V}\right)_T = \left(\frac{\partial P}{\partial T}\right)_V,\quad \left(\frac{\partial S}{\partial P}\right)_T = -\left(\frac{\partial V}{\partial T}\right)_P$$

Write the van der Waals equation of state.

$$\left(P + \frac{a}{V_m^{2}}\right)(V_m - b) = RT$$ where $a$ corrects for attraction and $b$ for molecular volume.

Define the compressibility factor $Z$ and its value for an ideal gas.

$$Z = \frac{PV_m}{RT}$$ For an ideal gas $Z = 1$; deviations measure non-ideality.

State the theorem of corresponding states (two-parameter form).

All fluids, when compared at the same reduced temperature $T_r = T/T_c$ and reduced pressure $P_r = P/P_c$, have approximately the same compressibility factor $Z$.

Define a residual property $M^{R}$.

$$M^{R} = M - M^{ig}$$ the difference between the actual molar property and the ideal-gas property at the same $T$ and $P$.

Write the residual Gibbs energy in terms of the fugacity coefficient.

$$\frac{G^{R}}{RT} = \ln \phi$$ where $\phi$ is the fugacity coefficient.

Express the residual enthalpy in terms of $Z$.

$$\frac{H^{R}}{RT} = -T\int_0^{P} \left(\frac{\partial Z}{\partial T}\right)_P \frac{dP}{P}$$

Define a partial molar property $\bar{M}_i$ of species $i$ in a mixture.

$$\bar{M}_i = \left(\frac{\partial (nM)}{\partial n_i}\right)_{T,P,n_{j\neq i}}$$

State the summability relation linking partial molar properties to the mixture property.

$$M = \sum_i x_i \bar{M}_i$$

Write the Gibbs–Duhem equation at constant $T$ and $P$.

$$\sum_i x_i \, d\bar{M}_i = 0 \qquad (\text{const } T,P)$$

What is the partial molar Gibbs energy equal to, and why is it central in phase equilibrium?

It equals the chemical potential: $\bar{G}_i = \mu_i$. Equality of $\mu_i$ across phases is the criterion for phase equilibrium.

Define the fugacity $f_i$ of a pure species via its chemical potential.

$$\mu_i = \Gamma_i(T) + RT \ln f_i$$ Fugacity is an 'effective pressure'; for an ideal gas $f_i = P$.

Define the fugacity coefficient $\phi_i$ for a pure species and for a species in a mixture.

Pure: $\phi = f/P$. In mixture: $\hat{\phi}_i = \dfrac{\hat{f}_i}{x_i P}$, where $\hat{f}_i$ is the partial fugacity of $i$.

State the criterion for phase equilibrium in terms of fugacities for each species $i$.

$$\hat{f}_i^{\,\alpha} = \hat{f}_i^{\,\beta} \quad \text{for all species } i \text{ and all phases.}$$

Define an excess property $M^{E}$.

$$M^{E} = M - M^{id}$$ the difference between the actual mixture property and the ideal-solution value at the same $T$, $P$, and composition.

What is the excess Gibbs energy's relation to activity coefficients?

$$\frac{G^{E}}{RT} = \sum_i x_i \ln \gamma_i$$ and $\ln\gamma_i = \dfrac{\partial (nG^E/RT)}{\partial n_i}$.

For an ideal solution, what are the values of $G^{E}$, $H^{E}$, and $V^{E}$?

All zero: $G^{E}=H^{E}=V^{E}=0$. Ideal mixing has $\Delta H_{mix}=0$, $\Delta V_{mix}=0$, and $\gamma_i = 1$.

Define the activity coefficient $\gamma_i$ and its ideal-solution value.

$$\gamma_i = \frac{\hat{f}_i}{x_i f_i}$$ which measures deviation from ideal-solution (Lewis–Randall) behavior; $\gamma_i = 1$ for an ideal solution.

Write the two-parameter Margules equation for activity coefficients of a binary.

$$\ln\gamma_1 = x_2^{2}\left[A_{12} + 2(A_{21}-A_{12})x_1\right]$$ $$\ln\gamma_2 = x_1^{2}\left[A_{21} + 2(A_{12}-A_{21})x_2\right]$$

Write the van Laar activity coefficient model for a binary system.

$$\ln\gamma_1 = A\left(1 + \frac{A x_1}{B x_2}\right)^{-2}, \qquad \ln\gamma_2 = B\left(1 + \frac{B x_2}{A x_1}\right)^{-2}$$

State Raoult's law for vapor–liquid equilibrium and its assumptions.

$$y_i P = x_i P_i^{sat}$$ Valid for ideal liquid ($\gamma_i=1$) and ideal vapor; species chemically similar.

Write the modified Raoult's law including non-ideality of the liquid phase.

$$y_i P = x_i \gamma_i P_i^{sat}$$ where $\gamma_i$ corrects for liquid-phase non-ideality.

State Henry's law for a dilute solute $i$ in VLE.

$$\hat{f}_i = x_i H_i$$ (often $y_i P = x_i H_i$), valid as $x_i \to 0$, where $H_i$ is Henry's constant.

Distinguish positive and negative deviations from Raoult's law and their azeotrope types.

Positive deviation: $\gamma_i > 1$, like-interactions weaker, gives minimum-boiling azeotrope. Negative deviation: $\gamma_i < 1$, stronger unlike-interactions, gives maximum-boiling azeotrope.

Write the general chemical reaction equilibrium criterion in terms of Gibbs energy.

$$\left(\frac{\partial G}{\partial \xi}\right)_{T,P} = \sum_i \nu_i \mu_i = 0$$ Gibbs energy is minimized at equilibrium.

Relate the equilibrium constant $K$ to the standard Gibbs energy of reaction, and write $K$ in terms of activities.

$$\ln K = -\frac{\Delta G^{\circ}}{RT}, \qquad K = \prod_i (\hat{a}_i)^{\nu_i}$$ where $\hat a_i = \hat f_i / f_i^{\circ}$. The van 't Hoff relation gives $\dfrac{d\ln K}{dT} = \dfrac{\Delta H^{\circ}}{RT^{2}}$.