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Question:

State third law of thermodynamics. Write its mathematical expression. Discuss some important consequences of third law.

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Question:

Define efficiency of a Carnot engine. A Carnot engine has an efficiency of 50%

when its sink temperature is at 270C. Calculate the source temperature for

increasing its efficiency to 60%.

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Question:

Derive an expression for the work done in an isothermal process of an ideal gas.

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Question:

What is an adiabatic index? Using the first law of thermodynamics, show tha

$TV-1=K,$ when one mole of an ideal gas is made to undergo quasi-static adiabatic expansion.

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Question:

For a $pVT-$ system, show that:

$\frac{dV}{V}=\alpha dT-\beta Tdp$

where $\beta T$ is the isothermal compressibility and  is isobaric coefficient of volume expansion.

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Question:

State Zeroth law of thermodynamics. Discuss how this law introduces the concept of temperature. Write parametric as well as exact equation of state for one mole of a real gas and paramagnetic substance.

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Question:

What are Intensive and extensive variables. Write two examples of each. List the

intensive and extensive variables required to specify the thermodynamic systems

(i) paramagnetic solid and (ii) stretched wire.

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Question:

What is Brownian motion? Write any four characteristics of Brownian motion.

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Question:

Define mean free path of the molecules of a gas. Show that it is equal to $\frac{1}{n\pi d^{2}}$

under Zeroth order approximation.

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Question:

The expression for the number of molecules in a Maxwellian gas having speeds in the range v to v + dv is given by

$dN_{v}=4\pi N\left ( \frac{m}{2\pi KBT} \right )^{3/2} V^{2}exp[-(\frac{mv^{2}}{2KBT})]dv$

Obtain an expression of average speed and root mean square speed of a molecule.

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Question:

Write the assumptions of kinetic theory of gases. Derive the following expression of the pressure exerted by an ideal gas:

$P=\frac{1}{3}mn\bar{v}^{2}$

Also, using this expression deduce Avogadro’s law. What is the kinetic interpretation temperature?

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Question:

For the function:

$f(x,y)=x^{3}+xy-2y^{2},$

Find the polynomial given by:

$f_{xx}(1,2)(x-2)^{2}+f_{xx}\left ( x-2 \right )(y-1)+f_{yy}(1,2)(y-1)^{2}$

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Question:

Show that the limit of the function $f(x,y)$ exists at the origin, where:

$f(x,y)$ $\left\{\begin{matrix} xcos\frac{1}{2} +ycos\frac{1}{x},(x,y)\neq & (0,0) & \\ 0& ,\ (x,y)=(0,0)& \end{matrix}\right.$

Do the repeated limits of $f(x,y)$ exist? Justify your answer.

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Question:

Solve the differential equation:

$(2xe^{y}y^{4}+2xy^{3}y)dx+(x^{2}y^{4}e^{y}-x^{2}y^{2}-3x)dy=0$

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Question:

Using the method of variation of parameters, solve the differential equation:

$\frac{d^{2}y}{dx^{2}}+y=sec^{3}\ x$

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Question:

Using Lagrange’s method, solve the differential equation:

$(x^{2}-y^{2}-z^{2})p+2xq=2xz$

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Question:

Find the envelope and the characteristic curves of the family of curves:

$x^{2}\left ( y-c\right )^{2}+z^{2}=c^{2}\ cos^{2}\ \alpha ,$

c and αare constants.

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Question:

Transform the given equation to Clairaut’s form and hence find its general solution:

$xy(y-px)=x-py$

Also find its singular solution, if it exists.

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Question:

Find the differential equations of the space curve in which the two families of surfaces:

$u=x^{2+y^{2}}+3z=c_{1}$

and $v=zx+x^{2}+y^{2}=c_{2.}$

intersect.

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Question:

The rate of change of the price of a commodity is proportional to the difference between the demand D and the supply S. If

$D=\alpha -bp$ and $s=c\ sin\ \beta t,$ where a, and $\beta$ are constants, determine $p(t).$ It is given that at $t=0,p=p_{0.}$

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