GATE Chemical Engineering Process Calculations and Thermodynamics Flashcards

24 sample cards from the Process Calculations and Thermodynamics deck

Sampled from the end of the deck, so these are different cards from the ones shown on the syllabus page.

1. 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.

2. 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 $Z$.

3. 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 $P$.

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

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

5. Express the residual enthalpy in terms of $Z$.

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

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

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

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

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

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

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

9. 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.

10. 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 $f_i = P$.

11. 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 $i$.

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

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

13. 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.

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

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

15. 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 $\gamma_i = 1$.

16. 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.

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

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

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

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

19. 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.

20. 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.

21. 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.

22. 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.

23. 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.

24. 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 $\dfrac{d\ln K}{dT} = \dfrac{\Delta H^{\circ}}{RT^{2}}$.

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