Programme : Plot specific heat of solids Dulong-Petit Law Einstein Distribution function Debye distribution function for high temperature and low temperature and compare them for these two cases. For copper Debye temperature is 345 K plot plot Cv VS T graph for Dulong-Petit law, Einstein law and Debye law. Find range of temperature at which Einstein and Debye law approaches Dulong Petit law. For Diamond ᶿD = 2100 K, plot the variation of specific heat at low temperature

Formula used and Theory

CV = 3R
CV = (3R (hv/KT)2 exp (hv/KT))(exp(hv/KT)-1)2
CV = (9R (T/ᶿD)3 (X4exdx)/(ex-1)2
Here R = Universal Gas Constant (Joule mol / Kelvin)
          h = Plank’s constant
          K = Boltzmann’s constant
          T = Temperature
         ᶿD = Debye Temperature

Algorithm

Input all the values of constants such as of R, h, ᶿD, KB etc.
Now define an array for temperature ranges such that t=linspace(1,3000,1000).
Now define the function for
Now, plot the different curves using plot command.
Label the graph using commands such as xlabel, ylabel, title, legend etc.

Coding

scf
clc
clf()
clear
r=8.314
h=6.626e-34
K=1.38e-23
thetaD=345              //EINSTEIN temperature
v=(K*thetaD/(h))
t=linspace(1,3000,1000)
function y=Cv1(t)
    y=3*r              //Dulong petit law
endfunction
for i=1:1000
    Cv2(i)=3*r*(h*v/(K*t(i)))^2*(exp(h*v/(K*t(i)))/((exp(h*v/(K*t(i))))-1)^2)          //Einstein Law
end
plot(t,Cv1,'g*')
plot(t,Cv2,'b*')
t1=linspace(1,3000,1000)
function y1=f(x)
    y1=(((x.^4)*(exp(x)))/((exp(x)-1).^2))
endfunction
i=1
for T=1:1:3000
z(i)=T
x(i)=(h*v)/(K*T)
y(i)=9*r*((T/thetaD).^3)*(intg(0,(thetaD/T),f))
i=i+1
end
plot(z,y)
title('specific heat V/s temperature for Dulong Petit, Einstein Law and Debye Law','fontsize',5)
xlabel('Temperature(Kelvin)','fontsize',5)
ylabel('Specific Heat','fontsize',5)
xgrid()

scf
clc
clf()
clear
r=8.314
h=6.626e-34
K=1.38e-23
thetaD=2100              //EINSTEIN temperature
v=(K*thetaD/(h))
t=linspace(1,500,100)
function y=Cv1(t)
    y=3*r              //Dulong petit law
endfunction
for i=1:100
    Cv2(i)=3*r*((h*v/(K*t(i)))^2)*(exp(-(h*v/(K*t(i)))))          //Einstein Law
end
plot(t,Cv1,'g*')
plot(t,Cv2,'b*')
thetaD1=2100             //DEBYE temperature
T=linspace(1,500,100)
for i=1:100
    Cv(i)=(12/5)*(%pi).^4*r*(T(i)/thetaD1)^3
end
plot(T,Cv)
title('specific heat V/s temperature for Dulong Petit, Einstein Law and Debye Law','fontsize',5)
xlabel('Temperature(Kelvin)','fontsize',5)
ylabel('Specific Heat','fontsize',5)
xgrid()

Result

According to Dulong Petit law the molar specific heat capacity at constant volume is given by Cv=3R=24.9 J / mol K
According to Einstein’s theory each atom vibrates with same angular frequency wE and may be treated as an independent oscillator. It means that all 3 Na modes of oscillation of a solid have the same frequency.
According to Debye theory molar specific heat capacity of solid is given by
CV = 3R[12 (T/ᶿD)3 (X3dx)/(ex-1))-3(ᶿD/T)/(exp(ᶿD/T)-1)]
The term within the square bracket is termed as the Debye function
At high temperature T>>ᶿD
So exp(ᶿD/T)-1 = exp(ᶿD/T)
Hence integration reduces to
 (X3dx)/(ex-1)) =  (X2dx) = (1/3)(ᶿD/T)3
And the second term becomes equal to 3. This means at high temp the Debye function becomes unity and Cv=3R.