Question

1. Calculating the number of degrees of freedom (how many additional intensive variables must be specified to fix the system) from the phase rule for the following materials at equilibrium

(a) Pure liquid benzene (b) A mixture of ice and water only (c) A mixture of liquid benzene, benzene vapor, and helium gas

(d) A mixture of salt and water designed to achieve a specific vapor pressure

2. Explain what happen when you increase the temperature of liquid benzene and toluene mixture at a fixed pressure, using T-xy diagram.

3. Reynolds number is a dimensionless parameter defined for a fluid flowing in a pipe as Re = Du?/?, where D is a pipe diameter, u is fluid velocity, ? is fluid density and ? is fluid viscosity. What is the dimension of fluid viscosity in terms of the base dimension types?

Answer 1

1. Formula used for calculation of degree of freedom inthermodynamics is

F + P = 2 +C.........Eq.1

where F is the number of degrees of freedom, C is the number of components, P is the number of phases.

a. For pure Liquid benzene, C (number of component) = 1, P (number of phase)=1.

Using equation 1,

F+1=2+1

F=2:

Hence, number of degrees of freedom is 2.

b. For a mixture of ice and water, C (number of component) = 1, P (number of phase)=2.

Using equation 1,

F+2=2+1

F=1:

Hence, number of degrees of freedom is 1.

3. For a mixture of liquid benzene, benzene vapor, and helium gas

C (number of component) = 2, P (number of phase)=2.

Using equation 1,

F+2=2+2

F=2:

Hence, number of degrees of freedom is 2.

d. For a mixture of salt and water designed to achieve a specific vapor pressure,

C (number of component) = 2 (water & NaCl)

P (number of phase)=1 (As mixture are about to acheive specific vapor pressure, it means to homogenous solution (salt competely dissloved in water)

Using equation 1,

F+1=2+2

F=3:

Hence, number of degrees of freedom is 3.

2.  When mixture of liquid benzene and toluene is heated up at constant pressure then it will remain liquid until it reached its bubble point temperature (point b in plot). This is the boiling point temperature of this mixture ( 'a' point composition). The vapor that boils off will not be the same composition of the liquid. Benzene is more volatile than toluene so there will be more benzene in the vapor mixture (point C). The two phases that are in equilibrium are at the same temperature and pressure, so they can be thought of as being connected through the two phase region by a horizontal line ( Line b-c). This horizontal line connecting the two equilibrium phases is called a tie line. If temperature continous increased upto point 'd' , dew point temperature, then more volatile liquid would be in vapor phase only.

3. Formula for Reyond number, Re = Du?/?.

Since Re is dimensionless number, its demsion are M⁰L⁰T⁰

Demsion for pipe diameter = M⁰L¹T⁰

Demsion for fluid density = M¹L^-3T⁰

In order to calculate the dimension of fluid viscosity, we use Rayleigh Method.

So, Re =function of (D,u,?,?)

Re^x = Du?/?^a , where a and x are exponents

M^xL^xT^x = [M⁰L¹T⁰ ] [M⁰L¹T^-1 ] [M¹L^-3T⁰ ] /[M^aL^aT^a ]

Equating the exponents to each other in terms of their respective fundamental units

For M , x = 0 = 0 +0+1-a;

a=1:

For L, x= 0 = 1+1-3-a:

a= -1:

For T, x= 0 = 0-1+-0-a:

a= -1:

Hence, Demsion for fluid viscoeity is M¹L^-1T^-1