GATE Examination

YEAR 2014

  • Q.1

    A spherical ball of benzoic acid (diameter = 1.5 cm) is submerged in a pool of still water. The solubility and diffusivity of benzoic acid in water are 0.03 kmol/m\(^3\) and 1.25 x 10\(^{-9}\) m\(^2\)/s respectively. Sherwood number is given as Sh = 2.0 + 0.6 Re\(^{0.5}\)Sc\(^{0.33}\). The initial rate of dissolution (in kmol/s) of benzoic acid approximately is

    (A) 

    \(3.54\times 10^{-11}\)

    (B) 

    \(3.54\times 10^{-12}\)

    (C) 

    \(3.54\times 10^{-13}\)

    (D) 

    \(3.54\times 10^{-14}\)

    Marks:2 Explanation
    • Answer:

      .

  • Q.2

    A wet solid of 100 kg is dried from a moisture content of 40 wt% to 10 wt%. The critical moisture content is 15 wt% and the equilibrium moisture content is negligible. All moisture contents are on dry basis. The falling rate is considered to be linear. It takes 5 hours to dry the material in the constant rate period. The duration (in hours) of the falling rate period is ___________

    Marks:2 Explanation
    • Answer:

      .

  • Q.3

    A brick wall of 20 cm thickness has thermal conductivity of 0.7 W m\(^{-1}\) K\(^{-1}\). An insulation of thermal conductivity 0.2 W m\(^{-1}\) K\(^{-1}\) is to be applied on one side of the wall, so that the heat transfer through the wall is reduced by 75%. The same temperature difference is maintained across the wall before and after applying the insulation. The required thickness (in cm) of the insulation is ____________

    Marks:2 Explanation
    • Answer:

      .

  • Q.4

    An oil with a flow rate of 1000 kg/h is to be cooled using water in a double-pipe counter-flow heat exchanger from a temperature of 70 °C to 40 °C. Water enters the exchanger at 25 °C and leaves at 40 °C. The specific heats of oil and water are 2 kJ kg\(^{-1}\) K\(^{-1}\) and 4.2 kJ kg\(^{-1}\) K\(^{-1}\), respectively. The overall heat transfer coefficient is 0.2 kW m\(^{-2}\) K\(^{-1}\). The minimum heat exchanger area (in m\(^{2}\)) required for this operation is ______________

    Marks:2 Explanation
    • Answer:

      .

  • Q.5

    Which ONE of the following is CORRECT for an ideal gas in a closed system?

     

    (A) 

    \((\frac{\partial U}{\partial V})_sV = nR(\frac{\partial U}{\partial S})_V\)

    (B) 

    \((\frac{\partial H}{\partial P})_sP = nR(\frac{\partial H}{\partial S})_P\)

    (C) 

    \((\frac{\partial U}{\partial V})_sV = nR(\frac{\partial H}{\partial S})_P\)

    (D) 

    \((\frac{\partial H}{\partial P})_SP = nR(\frac{\partial U}{\partial S})_V\)

    Marks:2 Explanation
    • Answer:

      .

  • Q.6

    A binary distillation column is operating with a mixed feed containing 20 mol% vapour. If the feed quality is changed to 80 mol% vapour, the change in the slope of the q-line is ________________

    Marks:2 Explanation
    • Answer:

      .

  • Q.7

    A homogeneous reaction (R→P) occurs in a batch reactor. The conversion of the reactant R is 67% after 10 minutes and 80% after 20 minutes. The rate equation for this reaction is

    (A) 

    \(-r_R = k\)

    (B) 

    \(-r_R = kC_R^2\)

    (C) 

    \(-r_R = kC_R^3\)

    (D) 

    \(-r_R = kC_R^{0.5}\)

    Marks:2 Explanation
    • Answer:

      .

  • Q.8

    A vapour phase catalytic reaction Q+R→S follows Rideal mechanism (R and S are not adsorbed). Initially, the mixture contains only the reactants in equimolar ratio. The surface reaction step is rate controlling. With constants a and b, the initial rate of reaction(\(-r_o\)) in terms of total pressure (\(P_T\)) is given by

    (A) 

    \(-r_o = \frac{aP_T}{1+bP_T}\)

    (B) 

    \(-r_o = \frac{aP_T}{1+bP_T^2}\)

    (C) 

    \(-r_o = \frac{aP_T^2}{1+bP_T}\)

    (D) 

    \(-r_o = \frac{aP_T^2}{1+bP_T^2}\)

    Marks:2 Explanation
    • Answer:

      .

  • Q.9

    A vapour phase catalytic reaction Q+R→S follows Rideal mechanism (R and S are not adsorbed). Initially, the mixture contains only the reactants in equimolar ratio. The surface reaction step is rate controlling. With constants a and b, the initial rate of reaction(\(-r_o\)) in terms of total pressure (\(P_T\)) is given by

    (A) 

    \(-r_o = \frac{aP_T}{1+bP_T}\)

    (B) 

    \(-r_o = \frac{aP_T}{1+bP_T^2}\)

    (C) 

    \(-r_o = \frac{aP_T^2}{1+bP_T}\)

    (D) 

    \(-r_o = \frac{aP_T^2}{1+bP_T^2}\)

    Marks:2 Explanation
    • Answer:

      .

  • Q.10

    An incompressible fluid is flowing through a contraction section of length L and has a 1-D (x-direction) steady state velocity distribution, \(u=u_o(1+\frac{2x}{L})\) . If u\(_o\) = 2 m/s and L = 3 m, the convective acceleration (in m/s\(^2\)) of the fluid at L is _____________

    Marks:2 Explanation
    • Answer:

      .

  • Q.11

    Match the following

    Group 1 Group 2
    P. Tank inSeries Model I. Non-isothermal reaction
    Q. Liquid-Liquid extraction II. Mixer settler
    R. Optimum temperature progression III. Solid catalyzed reaction
    S. Thiele modulus IV. Solid catalyzed reaction

     

    (A) 

    P─II, Q─IV, R─I, S─III

    (B) 

    P─I, Q─II, R─III, S─IV

    (C) 

    P─III, Q─I, R─II, S─IV

    (D) 

    P─III, Q─II, R─I, S─IV

    Marks:2 Explanation
    • Answer:

      .

  • Q.12

    Two elemental gases (A and B) are reacting to form a liquid (C) in a steady state process as per the reactionA+B→C. The single-pass conversion of the reaction is only 20% and hence recycle is used. The product is separated completely in pure form. The fresh feed has 49 mol% of A and B each along with 2 mol% impurities. The maximum allowable impurities in the recycle stream is 20 mol%. The amount of purge stream (in moles) per 100 moles of the fresh feed is ___________

    Marks:2 Explanation
    • Answer:

      .

  • Q.13

    Carbon monoxide (CO) is burnt in presence of 200% excess pure oxygen and the flame temperature achieved is 2298 K. The inlet streams are at 25 °C. The standard heat of formation (at 25 °C) of CO and CO\(_2\) are ─110 kJ mol\(^{-1}\) and ─390 kJ mol\(^{-1}\), respectively. The heat capacities (in J mol\(^{-1}\) K\(^{-1}\)) of the components are

                         \(C_{p_{O_2}} = 25 + 14\times 10^{-3}T \)

                         \(C_{p_{CO_2}} = 25 + 42\times 10^{-3}T \)

    where, T is the temperature in K. The heat loss (in kJ) per mole of CO burnt is_____________

    Marks:2 Explanation
    • Answer:

      .

  • Q.14

    A cash flow of Rs. 12,000 per year is received at the end of each year (uniform periodic payment) for 7 consecutive years. The rate of interest is 9% per year compounded annually. The present worth (in Rs.) of such cash flow at time zero is __________

    Marks:2 Explanation
    • Answer:

      .

  • Q.15

    A cash flow of Rs. 12,000 per year is received at the end of each year (uniform periodic payment) for 7 consecutive years. The rate of interest is 9% per year compounded annually. The present worth (in Rs.) of such cash flow at time zero is __________

    Marks:2 Explanation
    • Answer:

      .

  • Q.16

    A polymer plant with a production capacity of 10,000 tons per year has an overall yield of 70%, on mass basis (kg of product per kg of raw material). The raw material costs Rs. 50,000 per ton. A process modification is proposed to increase the overall yield to 75% with an investment of Rs. 12.5 crore. In how many years can the invested amount be recovered with the additional profit? ___________

    Marks:2 Explanation
    • Answer:

      .

  • Q.17

    A step change of magnitude 2 is introduced into a system having the following transfer function

                                              \(G(s) = {2\over s^2 \space + \space 2s \space +\space 4}\)

    The percent overshoot is ____________

    Marks:2 Explanation
    • Answer:

      .

  • Q.18

    Given below is a simplified block diagram of a feedforward control system.

                                                        

    The transfer function of the process is \(G_p = {5\over s+1}\) and the disturbance transfer function is \(G_d = {1\over s^2 \space + \space 2s \space + \space 1}\). The transfer function of the PERFECT feedforward controller, \(G_f(s)\) is

    (A) 

    \(-5\over (s+1)^3\)

    (B) 

    \(-5\over (s+1)\)

    (C) 

    \(-1\over 5(s+1)\)

    (D) 

    \(-5(s+1)\)

    Marks:2 Explanation
    • Answer:

      .

  • Q.19

    In a steady incompressible flow, the velocity distribution is given by \(\bar{V} = 3x\hat{i} \space - \space Py\hat{j} \space + \space 5z\hat{k}\) , where, V is in m/s and x, y, and z are in m. In order to satisfy the mass conservation, the value of the constantP (in s\(^{-1}\)) is __________

    Marks:2 Explanation
    • Answer:

      .

  • Q.20

    Match the following

                     

    Group 1 Group 2
    P. Turbulence I. Reciprocating Pump
    Q. NPSH II. Packed bed
    R. Ergun equation III. Fluctuating velocity
    S. Rotameter  IV. Impeller
    T. Power number  V. Vena contracta

     

    (A) 

    P─III, R─II, T─IV

    (B) 

    Q─V, R─II, S─III

    (C) 

    Q─V, R─II, S─III

    (D) 

    Q─III, S─V, T─IV

    Marks:2 Explanation
    • Answer:

      .

  • Q.21

    In a steady and incompressible flow of a fluid (density = 1.25 kg m\(^{-3}\)), the difference between stagnation and static pressures at the same location in the flow is 30 mm of mercury (density = 13600 kg m\(^{-3}\)). Considering gravitational acceleration as 10 m s\(^{-2}\), the fluid speed (in m s\(^{-1}\)) is ______________

    Marks:2 Explanation
    • Answer:

      The barometric head should be equal to velocity head i.e.

      \(\rho_{merc ury}gh=\frac{\rho_{fluid}V^2}{2}\)

      \(\implies V^2=\frac{2\times 13600\times 10\times0.03}{1.25}=6528\)

      \(\implies V=\sqrt {6528}=80.78m/s\)

        

  • Q.22

    In a closed system, the isentropic expansion of an ideal gas with constant specific heats is represented by

     

    (A) 

    (B) 

    (C) 

    (D) 

    Marks:1 Explanation
    • Answer:

      .Isentropic expansion follows the following equation:

      \(PV^{\gamma} = C\) ; where C  is a constant.

      Taking ln both sides we get:

      \(ln(PV^{\gamma}) = lnC\)

      \(=> lnP + lnV^{\gamma} = lnC\)

      \(=> lnP = -lnV^{\gamma} + lnC\)

      \(=> lnP = -\gamma \space lnV + lnC\)  ................(1) 

      Equation (1) can be compared with equation y = mx + c, where m = \(-\gamma\) and c = lnC.

      So. curve will be as below:

  • Q.23

    Match the following

    Group 1 Group 2
    P. \((\frac{\partial G}{\partial n_i})_{T,P,n_{j \neq 1}}\) I. Arrhenius equation
    Q. \((\frac{\partial G}{\partial n_i})_{S,V,n_{j \neq 1}}\) II. Reaction Equilibrium constant
    R. \(exp(\frac{-\bigtriangleup G^o_{reaction}}{RT})\) III. Chemical Potential
    S.\(\sum (n_id\mu_i)_{T,P} = 0\) IV. Gibbs-Duhem equation

     

    (A) 

    Q─III, R─I, S─II

    (B) 

    Q─III, R─II, S─IV

    (C) 

    P─III, R─II, S─IV

    (D) 

    P─III, R─IV, S─I

    Marks:1 Explanation
    • Answer:

      P. \((\frac{\partial G}{\partial n_i})_{T,P,n_{j \neq 1}}\)            = III. Chemical Potential

      R. \(exp(\frac{-\bigtriangleup G^o_{reaction}}{RT})\)      = II. Reaction Equilibrium constant

      S.\(\sum (n_id\mu_i)_{T,P} = 0\)    = IV. Gibbs-Duhem equation

  • Q.24

    In order to achieve the same conversion under identical reaction conditions and feed flow rate for a non-autocatalytic reaction of positive order, the volume of an ideal CSTR is

    (A) 

    always greater than that of an ideal PFR

    (B) 

    always smaller than that of an ideal PFR

    (C) 

    same as that of an ideal PFR

    (D) 

    smaller than that of an ideal PFR only for first order reaction

    Marks:1 Explanation
    • Answer:

      .

  • Q.25

    Integral of the time-weighted absolute error (ITAE) is expressed as

    (A) 

    \(\int^\infty_0 \frac{|\varepsilon(t)|}{t^2}dt\)

    (B) 

    \(\int^\infty_0 \frac{|\varepsilon(t)|}{t}dt\)

    (C) 

    \(\int^\infty_0 t{|\varepsilon(t)|}dt\)

    (D) 

    \(\int^\infty_0 t^2{|\varepsilon(t)|}dt\)

    Marks:1 Explanation
    • Answer:

      .

  • Q.26

    A unit IMPULSE response of a first order system with time constantτand steady state gain \(K_p\) is given by

    (A) 

    \(\frac{1}{K_p\tau}e^{t/\tau}\)

    (B) 

    \({K_p}e^{-t/\tau}\)

    (C) 

    \({K_p\tau}e^{-t/\tau}\)

    (D) 

    \(\frac{K_p}{\tau}e^{-t/\tau}\)

    Marks:1 Explanation
    • Answer:

      .

  • Q.27

    In a completely opaque medium, if 50% of the incident monochromatic radiation is absorbed, then which of the following statements are CORRECT?

    P. 50% of the incident radiation is reflected

    Q. 25% of the incident radiation is reflected

    R. 25% of the incident radiation is transmitted

    S. No incident radiation is transmitted

    (A) 

    P and S only

    (B) 

    Q and R only

    (C) 

    P and Q only

    (D) 

    R and S only

    Marks:1 Explanation
    • Answer:

      .

  • Q.28

    In case of a pressure driven laminar flow of a Newtonian fluid of viscosity (μ) through a horizontal circular pipe, the velocity of the fluid is proportional to

    (A) 

    μ

    (B) 

    \(\mu^{0.5}\)

    (C) 

    \(\mu^{-1}\)

    (D) 

    \(\mu^{-0.5}\)

    Marks:1 Explanation
    • Answer:

      .

  • Q.29

    Which of the following statements are CORRECT?

    P. For a rheopectic fluid, the apparent viscosity increases with time under a constant applied
    shear stress

    Q. For a pseudoplastic fluid, the apparent viscosity decreases with time under a constant applied
    shear stress

    R. For a Bingham plastic, the apparent viscosity increases exponentially with the deformation rate  

    S. For a dilatant fluid, the apparent viscosity increases with increasing deformation rate

    (A) 

    P and Q only

    (B) 

    Q and R only

    (C) 

    R and S only

    (D) 

    P and S only

    Marks:1 Explanation
    • Answer:

      .

  • Q.30

    Assume that an ordinary mercury-in-glass thermometer follows first order dynamics with a time constant of 10 s. It is at a steady state temperature of 0 °C. At time t = 0, the thermometer is suddenly immersed in a constant temperature bath at 100 °C. The time required (in s) for the thermometer to read 95 °C, approximately is

    (A) 

    60

    (B) 

    40

    (C) 

    30

    (D) 

    20

    Marks:1 Explanation
    • Answer:

      .

  • Q.31

    Packed towers are preferred for gas-liquid mass transfer operations with foaming liquids because

    (A) 

    in packed towers, high liquid to gas ratios are best handled

    (B) 

    in packed towers, continuous contact of gas and liquid takes place

    (C) 

    packed towers are packed with random packings

    (D) 

    in packed towers, the gas is not bubbled through the liquid pool

    Marks:1 Explanation
    • Answer:

      .

  • Q.32

    A spherical storage vessel is quarter–filled with toluene. The diameter of the vent at the top of the vessel is 1/20\(_{th}\) of the diameter of the vessel. Under the steady state condition, the diffusive flux of toluene is maximum at

    (A) 

    the surface of the liquid

    (B) 

    the mid-plane of the vessel

    (C) 

    the vent

    (D) 

    a distance 20 times the diameter of the vent away from the vent

    Marks:1 Explanation
    • Answer:

      .

  • Q.33

    In order to produce fine solid particles between 5 and 10 μm, the appropriate size reducing equipment is

    (A) 

    fluid energy mill

    (B) 

    hammer mill

    (C) 

    jaw crusher

    (D) 

    smooth roll crusher

    Marks:1 Explanation
    • Answer:

      .

  • Q.34

    Slurries are most conveniently pumped by a

    (A) 

    syringe pump

    (B) 

    diaphragm pump

    (C) 

    vacuum pump

    (D) 

    gear pump

    Marks:1 Explanation
    • Answer:

      .

  • Q.35

    Assuming the mass transfer coefficients in the gas and the liquid phases are comparable, the absorption of CO\(_2\) from reformer gas (CO\(_2\)+H\(_2\)) into an aqueous solution of diethanolamine is controlled by

    (A) 

    gas phase resistance

    (B) 

    liquid phase resistance

    (C) 

    both gas and liquid phase resistances

    (D) 

    composition of the reformer gas

    Marks:1 Explanation
    • Answer:

      .

  • Q.36

    Which ONE of the following statements is CORRECT for the surface renewal theory?

    (A) 

    Mass transfer takes place at steady state

    (B) 

    Mass transfer takes place at unsteady state

    (C) 

    Contact time is same for all the liquid elements

    (D) 

    Mass transfer depends only on the film resistance

    Marks:1 Explanation
    • Answer:

      .

  • Q.37

    Steam economy of a multiple effect evaporator system is defined as

    (A) 

    kilogram of steam used per hour

    (B) 

    kilogram of steam consumed in all the effects for each kilogram of steam fed

    (C) 

    kilogram of steam used in all the effects for each kilogram of water vaporized per hour

    (D) 

    kilogram of water vaporized from all the effects for each kilogram of steam fed to the first

    Marks:1 Explanation
    • Answer:

      .

  • Q.38

    Decomposition efficiency (η\(_D\)) of an electrolytic cell used for producing NaOH is defined as

    (A) 

    η\(_D\)= (grams of NaOH produced / grams of NaCl decomposed) x 100

    (B) 

    η\(_D\)= (grams of NaOH produced / grams of NaCl charged) x 100

    (C) 

    η\(_D\)= (gram equivalents of NaOH produced / gram equivalents of NaCl charged) x 100

    (D) 

    η\(_D\)= (theoretical current to produce one gram equivalent / actual current to produce one gram
    equivalent) x 100

    Marks:1 Explanation
    • Answer:

      .

  • Q.39

    The vessel dispersion number for an ideal CSTR is

    (A) 

    ─1

    (B) 

    0

    (C) 

    1

    (D) 

    \(\infty\)

    Marks:1 Explanation
    • Answer:

      .

  • Q.40

    Catalytic cracking is

    (A) 

    a hydrogen addition process

    (B) 

    a carbon rejection process

    (C) 

    an exothermic process

    (D) 

    a coking process

    Marks:1 Explanation
    • Answer:

      .

  • Q.41

    Which ONE of the following statements is CORRECT?

    (A) 

    The major components of biodiesel are triglycerides

    (B) 

    Biodiesel is essentially a mixture of ethyl esters

    (C) 

    Biodiesel is highly aromatic

    (D) 

    Biodiesel has a very low aniline point

    Marks:1 Explanation
    • Answer:

      .

  • Q.42

    Consider the following differential equation

    \(\frac{dy}{dx} = x \space + \space ln(y) \space ; \space y=2 \space at \space x=0\)

    The solution of this equation at x=0.4 using Euler method with a step size of h=0.2 is___________

    Marks:2 Explanation
    • Answer:

      .

  • Q.43

    The integrating factor for the differential equation

    \(\frac{dy}{dx} \space - \space \frac{y}{1+x} = (1+x) \space is\)

    (A) 

    \(\frac{1}{1+x}\)

    (B) 

    (1+x)

    (C) 

    x(1+x)

    (D) 

    \(\frac{x}{1+x}\)

    Marks:2 Explanation
    • Answer:

      .

  • Q.44

    The differential equation \(\frac{d^2y}{dx^2} \space + \space x^2\frac{dy}{dx} \space + \space x^3y = e^x \) is a

    (A) 

    non-linear differential equation of first degree

    (B) 

    linear differential equation of first degree

    (C) 

    linear differential equation of second degree

    (D) 

    non-linear differential equation of second degree

    Marks:2 Explanation
    • Answer:

      .

  • Q.45

    Consider the following two normal distributions

    \(f_1(x) = exp (-\pi x^2)\)

    \(f_2(x) = \frac{1}{2\pi}exp\{-\frac{1}{4\pi}(x^2+2x+1)\}\)

    If μ and σ denote the mean and standard deviation, respectively, then

    (A) 

    \(\mu_1 < \mu_2 \space and \space \sigma_1^2 < \sigma_2^2\)

    (B) 

    \(\mu_1 < \mu_2 \space and \space \sigma_1^2 > \sigma_2^2\)

    (C) 

    \(\mu_1 > \mu_2 \space and \space \sigma_1^2 < \sigma_2^2\)

    (D) 

    \(\mu_1 > \mu_2 \space and \space \sigma_1^2 > \sigma_2^2\)

    Marks:2 Explanation
    • Answer:

      .

  • Q.46

    In rolling of two fair dice, the outcome of an experiment is considered to be the sum of the numbers appearing on the dice. The probability is highest for the outcome of ____________

    Marks:2 Explanation
    • Answer:

      .

  • Q.47

    Gradient of a scalar variable is always

    (A) 

    a vector

    (B) 

    a scalar

    (C) 

    a dot product

    (D) 

    zero

    Marks:1 Explanation
    • Answer:

      The gradient operator \(\bigtriangledown\) is mathematically defined as \((\frac{\partial }{\partial x}\hat{i} + \frac{\partial }{\partial j}\hat{j} + \frac{\partial }{\partial z}\hat{k})\)

      So, for any scalar function F, \(\bigtriangledown F\) will always be vector.

  • Q.48

    For the time domain function \(f(t) = t^2,\)which ONE of the following is the Laplace transform of \(\int^t_0 f(t)dt?\)

    (A) 

    \(3\over s^4\)

    (B) 

    \(1\over 4s^2\)

    (C) 

    \(2\over s^3\)

    (D) 

    \(2\over s^4\)

    Marks:1 Explanation
    • Answer:

      The Laplace transform of time function of the type t\(^n\) is \(\frac{n!}{s^{n+1}}\), also if F(s) is the Laplace transform of any time domain function f(t) then the Laplace transform of \(\int{f(t)}dt \) is \(\frac{F(s)}{s}\).

      Here, f(t) = t\(^2\).

      F(s) = \(\frac{2!}{s^{2+1}} = \frac{2}{s^3}\)

      Then Laplace transorm of \(\int{t^2}dt \) is \(\frac{F(s)}{s}\) = \(\frac{2}{s^4}\)

  • Q.49

    If f*(x) is the complex conjugate of f(x) = cos(x) + isin(x), then for real a and b, \(\int^b_a f^*(x)f(x)dx\) is ALWAYS

    (A) 

    positive

    (B) 

    negative

    (C) 

    real

    (D) 

    imaginary

    Marks:1 Explanation
    • Answer:

      .f(x) = cos(x) + i sin(x)

      So, f*(x) = cos(x) - i sin(x)

      f(x) \(\times\) f*(x) = cos\(^2\)x + sin\(^2\)x = 1

      Now, \(\int^b_a f^*(x)f(x)dx\) = \(\int^b_a 1dx\) = \(x|^b_a\) = b - a

      Since, b and a real numbers so b - a will also be real.

      Therefore,  \(\int^b_a f^*(x)f(x)dx\) is always a real number.

  • Q.50

    If f(x) is a real and continuous function of x, the Taylor series expansion of f(x) about its minima will NEVER have a term containing

    (A) 

    first derivative

    (B) 

    second derivative

    (C) 

    third derivative

    (D) 

    any higher derivative

    Marks:1 Explanation
    • Answer:

      .We know that at minima or maxima of the curves the slope of the tangent to the curve is 0, which means for function f(x), f'(x) will be 0.

      Also, for finding the maxima and minima of the function f(x) we first find the point where f'(x) is zero, that point is suposed to be the maxima or minima depending upon the second derivative at that point. So, again going by this logic we can also see that first derivative of the function is 0.

      So, Taylor expansion of function f(x) will not contain the first derivative. 

  • Q.51

    From the following list, identify the properties which are equal in both vapour and liquid phases at equilibrium

    P. Density

    Q. Temperature

    R. Chemical potential

    S. Enthalpy

    (A) 

    P and Q only

    (B) 

    Q and R only

    (C) 

    R and S only

    (D) 

    P and S only

    Marks:1 Explanation
    • Answer:

      At equilibrium density cannot be equal in liquid and vapor phases as if they become equal then there will be no distiction between the phases and also vapors cannot have density equal to that of its corresponding liquid form.

      At equilibrium temperature can be equal if both phases of liquid and vapor as thermal equilibrium is also an equilibrium.

      Chemical potential is the driving force for chemical reaction (chemical/thermodynamic equilibrium) and is actually a composite quantity comprising of temperature, partial pressure and number of moles, so for thermodynamic equilibrium chemical potential need to be equal in both liquid and vapor phase.

      Enthalpy is the fumction of temperature and does depend upon the number of moles in the phase and C\(_p\)value so even with thermal equilibrium in effect enthalpy may not be equal as number of moles and C\(_p\)value for both phases may differ.

       

      So, we can concur that only statement Q and R is valid.