Chapter 9: Thermal Properties of Materials

Exa 9.1

In [1]:
from __future__ import division
import math
 # Python Code Ex9.1 Exception of Dulong-Petit law at room temp.:Page-303(2010)
 
 
 
 
#Variable declaration


h = 6.626e-034;                         # Planck's constant, joule second
k = 1.38e-023;                          # Boltzmann constant, joule/mol/kelvin
T = 300;                                # Room temperature, kelvin
f_Ag = 4.0e+012;            # Vibrational frequency for silver, cycles/second
f_Dia = 2.4e+013;          # Vibrational frequency for diamond, cycles/second



#Calculation

E_Ag = h*f_Ag;             # Vibrational Energy for silver, joule
E_Dia = h*f_Dia;            # Vibrational Energy for diamond, joule
E_th = k*T;                # Thermal energy at room temperature, joule



#Result

if (E_th > E_Ag and E_th < E_Dia):
  print"\nSince E_Ag < kT and E_Dia > kT, therefore," 
  print"\nSilver metal obeys the Dulong Petit law at "
  print "room temperature while diamond does not."

 

Since E_Ag < kT and E_Dia > kT, therefore,

Silver metal obeys the Dulong Petit law at 
room temperature while diamond does not.

Exa 9.2

In [2]:
from __future__ import division
import math
#Python Code Ex9.2 Specific heat of copper from Debye temp.:Page-311(2010)




#Variable declaration

h = 6.626e-034; # Planck's constant, joule second
k = 1.38e-023; # Boltzmann constant, joule/mol/kelvin
T = 30; # Given temperature, kelvin
N = 6.023e+023; # Avogadro's number
v_l = 4.76e+03; # Longitudinal velocity of lattice waves, m/s
v_t = 2.32e+03; # Tranverse velocity of lattice waves, 
rho = 8.9e+03; # Density of copper, kg per metre cube
A_Cu = 63.5; # Gram atomic mass of Cu, g



#Calculation

R = N*k; # Universal gas constant, joule/kelvin 
M = A_Cu*1e-03; # Mass of 1 mole of Cu-atoms, kg
V = M/rho; # Volume of copper, metre cube
# Debye temperature of copper, K
theta_D = (h/k)*((9*N)/((4*math.pi*V)*((1/v_l**3)+(2/v_t**3))))**(1/3);
 # Specific heat of copper, kJ/kmol/kelvin
C_v = 12/5*math.pi**4*R*(T/theta_D)**3; 



#Result

print"\nThe specific heat of copper =",round(C_v,2),"kJ/kmol/kelvin" 
 

The specific heat of copper = 1.33 kJ/kmol/kelvin

Exa 9.3

In [3]:
from __future__ import division
import math
#Python CodeEx9.3 Vibrational freq. and molar heat capacity of diamond:Page-312




#Variable declaration

h = 6.626e-034; # Planck's constant, joule second
k = 1.38e-023; # Boltzmann constant, joule/mol/kelvin
T = 10; # Given temperature, kelvin
N = 6.023e+023; # Avogadro's number
theta_D = 2230; # Debye temperature for diamond, kelvin



#Calculation

R = N*k; # Universal gas constant, joule/kelvin 
f_D = k*theta_D/h; # Debye frequency of diamond, hertz
# Specific heat of diamond, J/kmol/kelvin
C_v = 12/5*math.pi**4*R*1e+03*(T/theta_D)**3; 




#Result

print"\nThe highest possible vibrational frequency of diamond ="
print round((f_D*10**-13),2)*10**13,"per second" 
print"\nThe molar specific heat of diamond =",round(C_v,2),"J/kmol/kelvin"

The highest possible vibrational frequency of diamond =
4.64e+13 per second

The molar specific heat of diamond = 0.18 J/kmol/kelvin

Exa 9.4

In [4]:
from __future__ import division
import math
#Python Code Ex9.4Debye temperature of copper at low temperature:Page-312(2010)



#Variable declaration

k = 1.38e-023; # Boltzmann constant, joule/mol/kelvin
N = 6.023e+023; # Avogadro's number
C_vl = 4.6e-02; # Lattice specific heat, J/kmol/K



#Calculation

R = N*k; # Universal gas constant, joule/kelvin 
 # Lattice specific heat C_vl = Molar lattice specific heat, C_v
 # or 12/5*%math.pi**4*R/(5*theta_D**3) = C_vl
 # solving for theta_D, we have
# Debye temperature of copper at low temperature, K
theta_D = (12*math.pi**4*R*1e+03/(5*C_vl))**(1/3);



#Result

print"\nDebye temperature of copper at low temperature =",round(theta_D,2),"K"

Debye temperature of copper at low temperature = 348.27 K

Exa 9.5

In [5]:
from __future__ import division
import math
 # Python Code Ex9.5 Debye temperature for gold : Page-313(2010)
 
 
 
#Variable declaration


h = 6.626e-034; # Planck's constant, Js
k = 1.38e-023; # Boltzmann constant, joule/mol/kelvin
N = 6.023e+023; # Avogadro's number
v = 2100; # Velocity of sound in gold medium, m/s


#Calculation

M = 197e-03; # Gram atomic weight of gold, g
rho = 1.9e+04; # Density of gold, kg per metre cube
V = M/rho; # Volume of gold, metre cube
R = N*k; # Universal gas constant, joule/kelvin 
theta_D = h*v/k*(9*N/(12*math.pi*V))**(1/3); # Debye temperature for gold, K


#Result

print"\nDebye temperature of gold =",round(theta_D,2),"K"

Debye temperature of gold = 242.25 K

Exa 9.6

In [6]:
from __future__ import division
import math
 # Python Code Ex9.6 Heat transference into rock salt at
# low temperature: Page-313(2010)



#Variable declaration


A = 464;             # Atomic specific heat of rock salt, cal g/mol/kelvin
theta_D = 281;                          # Debye temperature of rock salt, K
delta_T = 10;              # Rise in temperature in each class interval, K





#Calculation

 # Define a function which returns lattice specific heat at constant volume

def Qa(t):
  return A*(t/theta_D)**3;
Q = 0; # Initialize heat accumulator to zero, cal
for t in range(10,50,10):
 #Calculate mean temperature of each class interval, K 
 mean_temp = (t + (t + 10))/2; 
 Q = Q + 2*delta_T*Qa(mean_temp);  # Acuumulate heat for each step
 
 
#Result

print"\nThe amount of heat required to raise the temperature of"
print" 2 gmol of Rock salt from 10K to 50K =",round(Q,2)," cal"

The amount of heat required to raise the temperature of
 2 gmol of Rock salt from 10K to 50K = 63.99  cal