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QUESTION: 1

A vessel of volume V = 30l contains ideal gas at temperature 0^{0}C. After a portion of the gas has been let out. the pressure in the vessel decreases by ΔP = 0.78 atm. Find the mass of the released gas. The gas density under the normal conditions is ρ = 1 .3 g/l. .

Solution:

Let m_{1} and m_{2} be the masses of the gas in vessel before and after the gas is released, the change in mass of released gas -

Δm =m_{1} - m _{2}

QUESTION: 2

An ideal gas whose pressure varies with volume according to the following relation P = P_{0} - aV^{2}. If we subject the gas to different pressures. What is the maximum temperature that the gas may attain?

Solution:

PV = nRT

given that P : P = P_{o} - aV^{2}

QUESTION: 3

A box containing 2 moles of a diatomic ideal gas at temperature T_{0} is connected to another identical box containing 2 moles of a monoatomic ideal gas at temperature 5T_{0}. There are no thermal losses and the heat capacity of the boxes and vibrational degree of freedom are negligible. Find the final temp, of the mixture

Solution:

Heat gained = heat lost

nCΔT=nC_{v}ΔT

QUESTION: 4

One mole of an ideal gas whose adiabatic exponent equals R undergoes a process in which the gas pressure relates to the temperature as P = aT^{α}. The work performed by the gas if its temperature gets an increment ΔT.

Solution:

p = aT^{α}

QUESTION: 5

The efficiency of a cycle consisting of two isobaric and two adiabatic lines, if the pressure changes n times within the cycle. The working substance is an ideal gas whose adiabatic exponent is equal to γ.

Solution:

QUESTION: 6

The heat required to convert 1 gm of water into steam under standard atm pressure if the entropy of water at the boiling point is 0.31 and for 100% steam at the steam temperature is 1.74 is:-

Solution:

ds = dQ/T

dQ = Tds

QUESTION: 7

If y = x cos x is a solution of an nth order linear differential equation

With real constant coefficients, then the least possible value of n is:-

Solution:

y = x cos x

y ^{1} = - x sinx + cosx

y^{n} = - sin x - x cos x - sin x

y^{n} + y = - 2 sin x...(i)

y^{n1} + y^{1} = - 2cosx

y +y^{n} = 2sinx...(ii)

(1)+(2) gives

y+ 2 y^{n} + y = 0

so, ⇒ n = 4

QUESTION: 8

The value of the integral

Solution:

using definition of gamma function

Here n = 4

QUESTION: 9

A monoatomic gas is described by the equation of state P(V - bn) = nRT Where b and R are constants and other quantities have their usual meanings. The maximum density (in makes per unit volume) to which this gas can be compressed is

Solution:

P (V- bn) = nRT

Differentiate equation with respect to V at constant T

for compression coefficient

for maximum K,

So. maximum density

QUESTION: 10

Consider the surface corresponding to the equation

A possible unit tangent to this surface at the point (1.2.-8) is

Solution:

The equation of surface is

= 4x^{2} + y^{2} + z = 0

So. gradient on the surface is

But unit tangent on the surface is along z-direction

QUESTION: 11

What are the eigen values of the given matrix:-

Solution:

QUESTION: 12

Which of the following matrices is Hermitian:-

Solution:

For the matrix A to Be Hermitian

A = ( A^{+})^{'
}

QUESTION: 13

An system at temperature T has three energy states 0, ± ε. The entropy of the system in the low temperature (T→0) and high temperature (T → ∝) limits are respectively,

Solution:

Using third law of thermodynamics entropy of closed system is moving toward to zero at zero tempS = KRf n zerature.

S_{T→0} = 0

We know partition function

Entropy S = K_{B}, log z

S = K_{B}ln z

QUESTION: 14

One mole of an ideal gas undergoes the cycle ACBA shown in the pV diagram below one of the curved lines in the cycle represents an isothermal change at temperature T, while the other represents and adiabatic change, the net heat gained by gas in this cycle is _____

Solution:

By the given conditions for the isothermal irreversible work

^{}

for the isothermal reversible 'work

-W=Q

⇒

Total work =

In isothermal process the net heat gain by the system used only for work. It is not contribute in internal e n erg y so the heat =

QUESTION: 15

An ideal gas at a temperature T is enclosed in a rigid container whose walls are initially at temperature T_{1}, where T_{1} < T. The walls are covered on the outside with perfect thermal insulation and the system is allowed to come to equilibrium. The pressure exerted by the gas on the walls of the container.

Solution:

We know that PV = nRT

P ∝ T

P_{1} ∝ T_{1}

PT_{1 }= P_{1}T

It means P > P_{1}

The pressure exerted by the gas on the walls of container is higher at the initial stage than at the final stage.

QUESTION: 16

Consider the CO molecule as a system of two point particles which has both translational and rotational degrees of freedom. Using classical statistical mechanics. the molar specific heat C_{v} of CO gas is given in form of the Boltzmann constant K_{B} by

Solution:

Translation degree of freedom = 3

Rotational degree of freedom = 2

QUESTION: 17

The vector is perpendicular to is perpendicular to Then the angle between

Solution:

..(i)

...(ii)

Subtracting

QUESTION: 18

In the temperature range 100 - 1000 C. the molar specific heat of a metal varies with temperature T (measured in degrees Celsius) according to the formula If 0.2 kg of the metal at 600 C is brought in thermal contact with 0.1 kg of the same metal at 300 c, the final equilibrium temperature in deg c, will be

Solution:

We know that at the thermal equilibrium

Heat absorbed = Heat released

QUESTION: 19

A monoatomic gas undergoes a process given by 2 du + 3dω = 0 Then the process is

Solution:

VT = constat

QUESTION: 20

An ideal gas in a Cylinder is Compressed adiabatic ally to one - third of its initial Volume. During this process 20 J work is done on the gas by compressing agent. Which of the following Statements is true for this case?

Solution:

dw = -20J

for adiabatic process

dO = du + dw = 0

du = - dw

du = 20J

QUESTION: 21

The ratio y = C_{P }/ C_{V }of H_{2}, gas is 7/5 at room temperature, and 5/3 at low temperatures (around 50 K). The reason for this change is

Solution:

We know at low temperature rotational motion of molecules is not consider so, total energy is due to the translatory or vibrational motion.

But at room temperature rotational motion is consider so

But at room temperature rotational motion is consider so

QUESTION: 22

Calculate the variation of C_{p} with pressure at constant temperature of a substance for which the equation of state is given the relation

Solution:

...(ii)

By the equation (1) and (2)

QUESTION: 23

f(x) is a periodic function of x with a period of 2π. In the interval -π<x<π, f(x) is given by

In the expansion of f(x) as a Fourier series of sine and cosine functions, the coefficient of cos 2x is

Solution:

( COS X = t )

QUESTION: 24

The volume of the portion of the cylinder x^{2} + y^{2} = 4 in the first octant between the lanes z = 0 and 3x - z = 0 is

Solution:

QUESTION: 25

For in isolated thermodynamical system, most probable, and root mean square values respectively of the molecular speeds of a gas room temperature being Maxwellian velocity distribution, then

Solution:

According to maxwell velocity distribution law

QUESTION: 26

For an isolated thermodynamical system P, V, T, U, S. and F represent the pressure. volume, temperature, internal energy, and free energy respectively. Then the following relation is true

Solution:

F = U - TS

dF = dl) - TdS - SdT

= (TdS - PdV) - TdS - SdT

= - PdV - SdT

QUESTION: 27

If z_{1} and z_{2} are two complex numbers then

Solution:

QUESTION: 28

The sublimation curve of solid ammonia is given by ln p = 23 - 3750/T and the vaporization curve of the liquid ammonia is given by In p = 19.5 -3050/T. where p is in mm of Hg and T is in K. The temperature of the triple point of ammonia is

Solution:

For sublimation of NH_{3}

...(i)

Fori vaporization of NH_{3}

At triple point

T = T_{r}

QUESTION: 29

The eigen values of the matrix are given by Which one of the following statement is NOT true?

Solution:

So, λ have following values

and two eigen values are complex. Ans. (C)

QUESTION: 30

Let is a solution of the Laplace equation then the vector field is

Solution:

So, is not solenoidal but irrotational.

*Multiple options can be correct

QUESTION: 31

Figure shows the P-V diagram of a cyclic process. If d6 is the heat energy supplied to the system, du is change in the internal energy of the system and dW is the work done by the system, then which of the following relations are correct-

Solution:

In a cyclic process, the system returns to its initial state. Hence the change in the internal energy du = 0. Therefore choice (B) is correct.

From the first law of thermodynamic

dθ = du + dW = dW

du = 0 ( Cyclic process )

*Multiple options can be correct

QUESTION: 32

The first law of thermodynamics is based on

Solution:

As we know that the first law of thermodynamics

d_{θ} = du + dW

That given equation states that the energy is conserve and equivalence of heat and work.

*Multiple options can be correct

QUESTION: 33

The following are the P-V diagrams for cyclic process for a gas. In Which of these process is heat absorbed by the gas?

Solution:

For cyclic process, ΔU = 0

Hence, if heart is absorbed by gas, work done by gas should be positive.

*Multiple options can be correct

QUESTION: 34

A particle moves along the curve X = 2t^{2}. y = t^{2} - 4t, z = 3t - 5 where t is time. Find the velocity and acceleration at t = 1 in the direction of

Solution:

The velocity component are given by

at t = 1

Now the component of velocity in the direction of

and the component of acceleration in the direction of

*Multiple options can be correct

QUESTION: 35

Which of the following are correct for the given matrix

Solution:

We know that eigen value are the roots of the equation then characteristic equation is

Hence eigenvectors corresponding to 0 and 2 are

*Multiple options can be correct

QUESTION: 36

Which of the following options are correct?

Solution:

is skew Hermitian

If

Inverse of a matrix exist, If matrix is non-singular.

*Multiple options can be correct

QUESTION: 37

A quantity of water is completely converted into steam by boiling.

Solution:

When water is boiling then it is completely converted in to steam. It is second order phase transition and we know at boiling point, the chemical potential in the vapour phase is less than that in liquid phase we know for specially water if pressure is increased than boiling temp would also increase and entropy of steam is greater than that of water Because disordered of molecules are increase when temp is rise.

*Multiple options can be correct

QUESTION: 38

The elasticity is defined as

Then which of the following sentences are Correct.

Solution:

*Multiple options can be correct

QUESTION: 39

One mole of an ideal gas is taken through the Cyclic process ABCA as shown in figure. The process BC is adiabatic and process CA is isothermal. γ = 1.5 then

Solution:

*Multiple options can be correct

QUESTION: 40

An ideal gas is taken from the state A (Pressure p. volume V) to the state B (pressure p/2, volume 2V) along a straight line path in the p - V diagram. Select the correct statements from the following

Solution:

Work done = Area under p-V graph

A_{1}>A_{2}

(B) In the given process p-V equation will be of a straight line with negative slope and positive intercept

(Here α and β are positive constants)

This is an equation of parabola in T and V.

T has some maximum value

⇒ T_{A }= T_{B}

We concluded that temperatures are same at A and B and in between temperature has a maximum value.

Therefore, in going from A to B, T will first increase to a maximum value and then decrease.

*Answer can only contain numeric values

QUESTION: 41

For the matrix the eigenvalue corresponding to the eigen vector is ____.

Solution:

Let, matrix

then let the given vector corresponding to the eigen value λ is

λ = 6

*Answer can only contain numeric values

QUESTION: 42

Two containers are maintained at the same temperature and are filled with ideal gases whose molecules have mass m_{1} and m_{2} respectively. The mean speed of molecules of the second gas in 10 times the rms speed of the molecules of the first gas. Find the ratio of m_{1}/m_{2} to the nearest integer.

Solution:

117-118

We know that

Average speed =

and root mean square speed =

Given that

*Answer can only contain numeric values

QUESTION: 43

N distinguishable particles are distributed among three states having energies E = 0, K_{B}T, 2K_{B}T respectively, If the total equilibrium energy of the system is 138.06 K_{a}T, find the number N of particles.

Solution:

We know that

Total energy

*Answer can only contain numeric values

QUESTION: 44

A certain amount of fluid with heat capacity C_{F} Joules /^{0}C is initially at a temperature O^{0}C, It is then brought into contact with a heat bath at a temperature of 100^{0}C, and the system is allowed to come into equilibrium. In this process, the entropy (in Joules / ^{0}C) of the universe changes by __________C_{F}.

Solution:

0.044

We know that entropy of the system

entropy of heat bath

Total entropy of the system

*Answer can only contain numeric values

QUESTION: 45

Consider the vector field F = where a is a constant. If then the value of a is ______ .

Solution:

(-a + 1)x + (1 - a) = 0

a = 1

*Answer can only contain numeric values

QUESTION: 46

Let C be the boundary of the region in the first quadrant by y = 1 - x^{2}, x = 0 and y = 0, oriented counter - clockwise. The value of

Solution:

Using Green theorem

=

*Answer can only contain numeric values

QUESTION: 47

Let C be the boundary of the triangle with vertices (0, 1, 0), (1,0, 0) and (2,1, 0). If then evaluate when C is traversed counter clockwise when viewed from above.

Solution:

According to stoke's theorem

For the surface bounded by triangle,

*Answer can only contain numeric values

QUESTION: 48

The number of values of λ for which the system of equations

has infinitely many solutions, is _______ .

Solution:

Coefficient matrix is given by

The determinant will be zero for two values of λ.

*Answer can only contain numeric values

QUESTION: 49

A thermally isolated container stores gas at 27.24^{0}C at one atmospheric pressure. Suddenly the pressure of the gas is increased to two atmospheric pressures. Assuming N_{2}, to behave as an ideal gas, estimate the change in temperature of the gas, in Celsius degrees (^{0}C).

Solution:

We know that

P_{1} = 1 atm

T_{1} = 300.24 k

P_{2} = 2 atm

T_{2} = ?

*Answer can only contain numeric values

QUESTION: 50

A one mole monoatomic gas and one mole of a diatomic gas are mixed in equal ratio, then find out the γ of mixture.

Solution:

We know that

similarly

So

*Answer can only contain numeric values

QUESTION: 51

Two glass bulbs of equal volumes are connected by a narrow tube and are filled with a gas at o^{0}C and a pressure of 76cm of Hg. One of the bulbs is then placed in water both maintained at 62^{0}C, what is the new value of pressure inside the bulbs (in cm of Hg)? (The volume of connecting tube is negligible.)

Solution:

PV = nRT

As total number of air molecules in the bulbs would remain constant, we have

As temperature of second bulb is uncharged, we have

*Answer can only contain numeric values

QUESTION: 52

5 kg of water at 0^{0}C is mixed with an equal mass of water at 80^{0}C. What is the change in entropy (in Cal/k). (Specific heat of water may be assumed to be equal to 1 kcal/kg ^{0}C between 0^{0}C at 80^{0}C.)?

Solution:

Let the final temperature of the mixture t^{0}C Then

5 x1 (t - 0)= 5x (80 - t)

t = 40^{0}°C = 313k

Now increase in entropy when the temperature of 5kg of water rises from 273 kto 313 k

Also decrease in entropy When the temperature of 5kg of water Falls from 80^{0}c (353k) to 313k

= 602.1 cal / k

*Answer can only contain numeric values

QUESTION: 53

A 42 kilowatt engine is operating between 227^{0}C and 177^{0}C, then the efficiency of engine is _____ .

Solution:

(i)

η = 0.1 = 10%

*Answer can only contain numeric values

QUESTION: 54

A carnot engine operating between two temperatures 727^{0}C and 27^{0}C is supplied heat energy at the rate of 500 joule/cycle. 60% of the work output is used to derive a refrigerator, which rejects heat to the surrounding at 27^{0}C. If the refrigerator removes 1050 Joule of heat per cycle from the low temperature reservoir, determine the temperature of the reservoir (in K).

Solution:

Efficiency

w = 0.7 x 500

= 350 J

θ_{1} + w = θ_{2}

work in put = 0.60x350 = 210 J

*Answer can only contain numeric values

QUESTION: 55

If then what is the value of c.

Solution:

y(x) = In (x+c)

y(0) = linc

c = e^{0 }⇒ c = 1

*Answer can only contain numeric values

QUESTION: 56

If 250g of Ni at 120 ^{0}C is dropped into 200g of water at 10 ^{0}C contained by a Calorimeter of 20 Cal / ^{0}C heat Capacity, what will be the final temperature of the mixture (in ^{0}C)? Given ( C_{N} = .106 k Cal / ^{0}c)

Solution:

Heat tost by N_{i} = heat gained by water + Calorimeter

Heat capacity of N_{i}, C_{N}= 0.106 k cal /^{0}c

So, 0.250 (0.106) (120^{0}-1) = [ (10.200) (1.00 + 0.020) x (t - 10^{0}c)

3.18 -0.07t = 0.220 t-2.20

0.2471 = 5.38

t = 22^{0}c

*Answer can only contain numeric values

QUESTION: 57

he diameter of ^{4}He atom is . One mole of the gas occupies 20 Ltr at 20 K. Calculate the mean free path of the molecules. (In nm)

Solution:

We know that

λ = 7.5 x 10^{-7} m

λ = 750 nm

*Answer can only contain numeric values

QUESTION: 58

Determine the mean free path in nitrogen given that the density of nitrogen at NTP = 1.2 x 10^{-3}g / cm^{3} and that its coefficient of viscosity is 1.7 x 10^{-4}g / cm^{3} per unit velocity gradient. 1 amu

Solution:

The mean free path

η= 1.7x10^{-4} dyne/cm^{2} per unit velocity gradient

*Answer can only contain numeric values

QUESTION: 59

Calculate the change in the boiling point of water (in K) when the pressure is increased from 1 to 10 atm.

Given - Specific volume of steam = 1677 cc / gm

Latent heat of steam = 540 cal / gm

boiling point of water at one atm pressure = 100^{0}C = 373 k

and pressure of one atm = 10^{6} dyne / cm^{2}

Solution:

dP = 10 -1 =9 atm = 9 x 10^{6}dyne / cm^{2}

T = Boiling point of water = 100^{0}c = 373 K

V_{2}= 1677 cc /gm

V_{1} = 1 cc /gm

V_{2} - V_{2} = 1676 cc/gm

L = 540 Cal/gm

= 540 x 4.2 x10^{7 }erg/gm

dt = 248.07 K

*Answer can only contain numeric values

QUESTION: 60

The Vapour pressure P (in mm of Hg) of Solid ammonia is given by

While that of Liquid ammonia is given by

If the Coordinate of triple point of ammonia is ( P.T) then calculate [P + T]?

Solution:

At triple point, the Vapour pressure of the Substance in each of three states is identical

AT triple point

T = 195.2 K

P = 44.87 mm at Hg

(P, T) = (44.87,195.2)

P + T =240

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