Using the partition function for an ideal gas made up of N indistinguishable particles, obtain Sackur-Tetrode equation.

Calculate the most probable speed, average speed and the root mean square speed for gas molecules at 300 K. The mass of gas molecule is 5 x 10 ^-26 kg and k_B = 1.38 x 10 ^-23JK ^-1.

random variable X has the following probability distribution:

Determine the value of the constant k and E(X).

One mole of an ideal gas expands isothermally to five times its initial volume. Calculate the entropy change in terms of R, the gas constant. 

A sonometer wire having cross-sectional area 0.85 x 10^-6m² is stretched between two rigid supports 1.2 m apart. A tension of 20 N is applied at its free end. If the temperature is reduced by 12°C, calculate the final tension in the wire. Take coefficient 1.5 x tof linear expansion (a) and isothermal Young's modulus (y) to be 1.5x10^-5 K ^-1 and 2.0 x 10¹¹ Nm ^-2,respectively.

The construction of a building may be delayed by a strike. Suppose that the probability that there will be a strike is 0.70, the probability that the construction

will be completed on time if there is no strike is 0.90 and the probability that the building will be completed on time if there is a strike is 0.40, what is the probability that the construction will be completed on time. 

Using Green’s Theorem evaluate the integral where C is a circle of radius 3 units centered at the origin.

Calculate the divergence of the vector function

Using the divergence theorem evaluate where  and S is the surface of the cube defined by 

Calculate the work done by a force  in moving a particle counter clockwise along the circle x²+y²=4 from the point (2, 0) to the point (0, -2).

Obtain a solution of the wave equation

Obtain the Fourier series expansion for the following periodic function which has a period of π:

Obtain the divergence and curl of the following vector field:

Evaluate .

b) Show that for any scalar field :

.

Reduce the following PDE to a set of three ODEs by the method of separation of variables.

Obtain a unit tangent vector to any point on the curve defined by the parameteric equations:

x = sin 3t; y = 2 cos 3t;z = 4t.

b) Obtain the directional derivative for a scalar field at the point (1,-2,-1) in the direction .

Obtain all the first and second order partial derivatives of the function:

f (x, y) = sin xy

Obtain a unit vector perpendicular to the plane of the vectors

 and 

b) For any four vectors   and   determine:

Show that the function

i) u(x,y)=  is a solution of the two-dimensional Laplace’s equation.

ii) u(x,t)=e^{-6x cos2x} is a solution of the one-dimensional heat equation.

Solve the following ODE using the power series method:

(x+2)y"+xy'-y=0