The constant pressure molar heat capacity of zinc is 25.40 J K-1 mol-1 at 298 K. Calculate the constant pressure specific heat capacity of zinc at this temperature. The molar mass of zinc is 65.37 g mol-1. 0.3886 K-1 kg-1 388.6 J K-1 kg-1 1.660 kJ K-1 kg-1 33.71 J K-1 kg-1
The Correct Answer and Explanation is:
To calculate the constant pressure specific heat capacity of zinc at 298 K, we use the relationship:
cp=Cp,mMc_p = \frac{C_{p,m}}{M}
Where:
- cpc_p is the specific heat capacity (J K⁻¹ kg⁻¹)
- Cp,mC_{p,m} is the molar heat capacity (25.40 J K⁻¹ mol⁻¹)
- MM is the molar mass of zinc (65.37 g mol⁻¹ or 0.06537 kg mol⁻¹)
cp=25.40 J K−1mol−10.06537 kg mol−1≈388.6 J K−1kg−1c_p = \frac{25.40\ \text{J K}^{-1} \text{mol}^{-1}}{0.06537\ \text{kg mol}^{-1}} \approx 388.6\ \text{J K}^{-1} \text{kg}^{-1}
Correct answer: 388.6 J K⁻¹ kg⁻¹
Explanation (about 300 words): This problem centers on translating a substance’s molar heat capacity into its specific heat capacity. The molar heat capacity tells us how much energy is required to raise the temperature of one mole of a substance by one Kelvin. However, specific heat capacity reflects how much energy is needed to raise the temperature of one kilogram of the material by the same amount.
To perform this conversion, you divide the molar heat capacity by the molar mass expressed in kilograms per mole. This ensures consistent units. Since the molar mass of zinc is given in grams per mole, you convert grams to kilograms by dividing by 1000.
Mathematically, this becomes:
25.400.06537=388.6 J K−1kg−1\frac{25.40}{0.06537} = 388.6\ \text{J K}^{-1} \text{kg}^{-1}
This value falls squarely among the provided multiple-choice answers, confirming that the calculation is correct.
Understanding this relationship is essential in thermodynamics, especially when analyzing energy transfer in materials. Scientists and engineers often use specific heat when working with physical samples measured by mass rather than by the number of moles. This kind of conversion enables them to relate theoretical values from tables to real-world systems such as metal heating in manufacturing or material selection in engineering.
In this case, zinc’s specific heat capacity indicates that it takes 388.6 joules of energy to raise the temperature of one kilogram of zinc by one Kelvin under constant pressure. This moderate value is characteristic of metals, which generally have lower heat capacities than non-metals due to the structure of metallic bonding.
