For aluminum, the heat capacity at constant volume Cv at 30 K is 0.81 J/mol-K, and the Debye temperature is 375 K. Estimate the specific heat
(a) At 50 K and
(b) At 425 K.
The Correct Answer and Explanation is :
To estimate the specific heat of aluminum at 50 K and 425 K, we use the Debye model, which describes the heat capacity of solids at different temperature ranges.
Debye Model Formula for Heat Capacity:
The Debye model gives the heat capacity at constant volume as:
[
C_V = 9Nk_B \left(\frac{T}{\Theta_D}\right)^3 \int_0^{\Theta_D/T} \frac{x^4 e^x}{(e^x -1)^2} dx
]
where:
- ( C_V ) = heat capacity at constant volume
- ( N ) = number of atoms per mole (Avogadro’s number)
- ( k_B ) = Boltzmann constant
- ( T ) = temperature
- ( \Theta_D ) = Debye temperature
- ( x = \frac{h \nu}{k_B T} ), integration variable
However, in practical calculations, we use approximations based on temperature ranges:
- For ( T \ll \Theta_D ) (low-temperature regime, ( T < 0.1\Theta_D )):
[
C_V \approx \frac{12\pi^4}{5} Nk_B \left(\frac{T}{\Theta_D}\right)^3
]
This follows the Debye (T^3) Law. - For ( T \gg \Theta_D ) (high-temperature regime, ( T > \Theta_D )):
[
C_V \approx 3Nk_B
]
This approaches the Dulong–Petit limit (~25 J/mol-K for aluminum).
(a) Specific Heat at 50 K
Since ( 50 K < \Theta_D ), we use the Debye approximation:
[
C_V(50K) = C_V(30K) \times \left(\frac{50}{30}\right)^3
]
[
C_V(50K) = 0.81 \times \left(\frac{50}{30}\right)^3
]
[
C_V(50K) \approx 3.74 J/mol-K
]
(b) Specific Heat at 425 K
Since ( 425 K > \Theta_D ), aluminum is in the classical regime, where the heat capacity approaches the Dulong–Petit law:
[
C_V(425K) \approx 3R = 24.9 J/mol-K
]
where ( R ) is the gas constant (( 8.314 J/mol-K )).
Final Answers:
- ( C_V(50K) \approx 3.74 J/mol-K )
- ( C_V(425K) \approx 24.9 J/mol-K ) (approaching the classical limit)
I’ll now generate an image to illustrate this.
Here is the generated diagram illustrating the heat capacity of aluminum as a function of temperature. It shows the two main regions: the low-temperature Debye ( T^3 ) regime and the high-temperature Dulong-Petit limit. Let me know if you need any modifications!
