The molar specific heat at constant pressure for a certain...

The molar specific heat at constant pressure for a certain gas is given by $C_{p}=a+b T+c T^{2}$, where $a=33.6 \mathrm{~J} / \mathrm{mol} \cdot \mathrm{K}$, $b=2.93 \times 10^{-3} \mathrm{~J} / \mathrm{mol} \cdot \mathrm{K}^{2}$, and $c=2.13 \times 10^{-5} \mathrm{~J} / \mathrm{mol} \cdot \mathrm{K}^{3}$. Find the entropy change when $2.00$ moles of this gas are heated from $20.0^{\circ} \mathrm{C}$ to $200^{\circ} \mathrm{C}$.

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Key Concepts

Temperature Conversion to Kelvin

In thermodynamic calculations, it is essential to use the absolute temperature scale (Kelvin) to avoid inconsistencies and to correctly apply the laws of thermodynamics. Temperature conversion ensures that the integration of thermodynamic quantities is done accurately.

Integration of Temperature-Dependent Specific Heat

When the specific heat capacity is a function of temperature, as in the given case, the entropy change must be determined by integrating the expression C?(T)/T with respect to temperature. This integration accounts for the variations in heat capacity over the temperature range of the process.

Entropy

Entropy is a measure of the disorder or randomness in a system, and in thermodynamics it is a state function that quantifies the dispersal of energy in a process. Changes in entropy can be calculated for processes by integrating the heat transfer divided by the temperature, particularly in reversible transformations.

Specific Heat Capacity at Constant Pressure

The specific heat capacity at constant pressure (C?) quantifies the amount of heat required to raise the temperature of one mole of a substance by one Kelvin while maintaining constant pressure. It is a key property in calculating the heat transfer and subsequent entropy changes in thermal processes.

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