17.4 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. Solution (a) For aluminum, Cv at 50 K may be approximated by Equation 17.2, since this temperature is significantly below the Debye temperature (375 K). The value of Cv at 30 K is given, and thus, we may compute the constant A as A = Cv T 3 = 0.81 J/mol-K (30 K) 3 = 3.00 ´ 10-5 J/mol-K 4 Therefore, at 50 K Cv = AT 3 = (3.00 ´ 10-5 J/mol-K 4 )(50 K) 3 = 3.75 J/mol-K and cv = (3.75 J/mol-K)(1 mol/26.98 g)(1000 g/kg) = 139 J/kg-K (b) Since 425 K is above the Debye temperature, a good approximation for C
The Correct Answer and Explanation is :
To estimate the specific heat capacity of aluminum at 50 K and 425 K, we use different approximations based on the relationship between the given temperature and the Debye temperature (Θ_D = 375 K).
(a) Specific Heat at 50 K
Since 50 K is much lower than Θ_D, we use the low-temperature approximation given by the Debye model:
[
C_v = A T^3
]
where ( A ) is determined from the given value at 30 K:
[
A = C_v \frac{T^3}{T^3} = (0.81 \text{ J/mol-K}) \times (30 K)^3
]
[
A = 3.00 \times 10^{-5} \text{ J/mol-K}^4
]
Now, we compute ( C_v ) at 50 K:
[
C_v = A T^3 = (3.00 \times 10^{-5} \text{ J/mol-K}^4) \times (50 K)^3
]
[
C_v = 3.75 \text{ J/mol-K}
]
The specific heat in J/kg-K is obtained by dividing by the molar mass of aluminum (26.98 g/mol):
[
c_v = \left( 3.75 \text{ J/mol-K} \right) \times \left( \frac{1 \text{ mol}}{26.98 \text{ g}} \right) \times \left( 1000 \text{ g/kg} \right)
]
[
c_v = 139 \text{ J/kg-K}
]
(b) Specific Heat at 425 K
For temperatures above Θ_D, the heat capacity follows the classical Dulong-Petit law:
[
C_v \approx 3R
]
where ( R = 8.314 \text{ J/mol-K} ), so:
[
C_v = 3 \times 8.314 = 24.94 \text{ J/mol-K}
]
The specific heat in J/kg-K:
[
c_v = \left( 24.94 \text{ J/mol-K} \right) \times \left( \frac{1 \text{ mol}}{26.98 \text{ g}} \right) \times \left( 1000 \text{ g/kg} \right)
]
[
c_v = 924 \text{ J/kg-K}
]
Explanation
- At low temperatures (T << Θ_D), the Debye model predicts that heat capacity follows the ( T^3 ) dependence.
- At high temperatures (T > Θ_D), atomic vibrations follow classical behavior, and the heat capacity approaches Dulong-Petit’s law with a nearly constant value of 3R.
- Using these approximations, we find that at 50 K, ( c_v = 139 ) J/kg-K, and at 425 K, ( c_v = 924 ) J/kg-K.
Now, I will generate an image illustrating the temperature dependence of specific heat.
Here is the scientific diagram illustrating the specific heat capacity of aluminum as a function of temperature. The graph follows the Debye model at low temperatures (T³ dependence) and levels off at high temperatures according to the Dulong-Petit law (3R). Key points such as 30K, 50K, 375K (Debye temperature), and 425K are marked. Let me know if you need any modifications!
