Question

Debye's theory of solids gives the heat capacity of a solid at temperature T to be p) Jo(e*-1)2 Write a Python function cv (T) that calculates Cv for a given value of the temperature, for a sample consisting of 1000 cubic centimeters of solid aluminum, which has a number density of -б.022 x 1028 m-3 and a Debye temperature of 428 K Use your function to make a graph of the heat capacity as a function of temperature from T-5K to T 500K scipy.integrate.quad () to evaluate the integral, rather than using Gaussian quadrature. The point is that scipy. integrate, quad() is a good general-purpose integrator that will automatically pick a numer- ical method suitable for many types of problem. Programming notes: . Put everything in SI units: V in m3, Кв in J/K, etc. For example, rho- 6, 022e28 # number density of aluminum (m^-3) Then Cv will come out in J/K. Be sure to label the axes of your plot including units python using spyder )

Answer 1

CODE

import math
from scipy import integrate
import matplotlib.pyplot as plt
def cV(T):
V = 0.001 # aluminum volume in cubic meter
rho = 6.022e28 # number density of aluminim in (m^-3)
Td = 428 #debye temperature
kb = 1.38064852e-28 # boltzmann constant in J/K
#function to be integerated
f = lambda x: (x**4)*(math.exp( x ))/(math.exp( x ) - 1)**(2)
INT = integrate.quad(f, 0, Td/T)
#Cv at T
value = 9*V*rho*kb*((T/Td)**3) * (INT[0] - INT[1])
return value

#heat capacity lst
cv = []
i = 0
T = list(range(5,301))
l = 301- 5
while(i < l):
cv.append(cV(T[i]))
i = i+ 1

#plotting
plt.plot(T, cv, 'ro')
plt.xlabel('Temperature (K)')
plt.ylabel('Cv (J/K)')
plt.title('Aluminiums Heat Capacity with Temperature')
plt.show()

SCREENSHOTS

In [24]: import math from scipy import integrate import matplotlib.pyplot as plt def cV(T): V -0.091 # aluminum volume in cubic meter rho 6.822e28 # number density of aluminim 1n (m^-3) 428 #debye temperature kb 1.38064852e-28 # boltzmann constant in J/K #function to be integerated f = lambda x: (x**4) * (math.expC x ))/(math . exp( x ) INT İntegrate . quad(f, θ, Td/T) #Cv at T value -9*V*rho kb ((T/Td)**3) *(INT[e] -INT[1]) return value -1)""(2) #heat capacity 1st CV F T-list(range(5,301)) 1 301-5 while(i 1): cv.append (cV(T[i])) #plotting plt.plot(T, cv, 'ro' plt.xlabel( Temperature (K)') pit . ylabel('Cv (J/K).) plt.title('Aluminiums Heat Capacity with Temperature') plt.show)

Aluminiums Heat Capacity with Temperature 0.025 0.020 0.015 0.010 0.005 0.000 250 0 50 100 150 Temperature (K) 200 300

Note: Please take care of the the indentation as it is lost while pasting

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