Answer:
The specific heat of copper is [tex]C= 392 J/kg\cdot ^o K[/tex]
Explanation:
From the question we are told that
The amount of energy contributed by each oscillating lattice site is [tex]E =3 kT[/tex]
The atomic mass of copper is [tex]M = 63.6 g/mol[/tex]
The atomic mass of aluminum is [tex]m_a = 27.0g/mol[/tex]
The specific heat of aluminum is [tex]c_a = 900 J/kg-K[/tex]
The objective of this solution is to obtain the specific heat of copper
Now specific heat can be defined as the heat required to raise the temperature of 1 kg of a substance by [tex]1 ^o K[/tex]
The general equation for specific heat is
[tex]C = \frac{dU}{dT}[/tex]
Where [tex]dT[/tex] is the change in temperature
[tex]dU[/tex] is the change in internal energy
The internal energy is mathematically evaluated as
[tex]U = 3nk_BT[/tex]
Where [tex]k_B[/tex] is the Boltzmann constant with a value of [tex]1.38*10^{-23} kg \cdot m^2 /s^2 \cdot ^o K[/tex]
T is the room temperature
n is the number of atoms in a substance
Generally number of atoms in mass of an element can be obtained using the mathematical operation
[tex]n = \frac{m}{M} * N_A[/tex]
Where [tex]N_A[/tex] is the Avogadro's number with a constant value of [tex]6.022*10^{23} / mol[/tex]
M is the atomic mass of the element
m actual mass of the element
So the number of atoms in 1 kg of copper is evaluated as
[tex]m = 1 kg = 1 kg * \frac{10000 g}{1kg } = 1000g[/tex]
The number of atom is
[tex]n = \frac{1000}{63.6} * (6.0*0^{23})[/tex]
[tex]= 9.46*10^{24} \ atoms[/tex]
Now substituting the equation for internal energy into the equation for specific heat
[tex]C = \frac{d}{dT} (3 n k_B T)[/tex]
[tex]=3nk_B[/tex]
Substituting values
[tex]C = 3 (9.46*10^{24} )(1.38 *10^{-23})[/tex]
[tex]C= 392 J/kg\cdot ^o K[/tex]
