The following data show how the standard molar...

The following data show how the standard molar constant-pressure heat capacity of sulfur dioxide varies with temperature: $$\begin{array}{lllllll} T / \mathrm{K} & 300 & 500 & 700 & 900 & 1100 & 1300 & 1500 \\ C_{p, m}^{\circ} /\left(\mathrm{JK}^{-1} \mathrm{mol}^{-1}\right) & 39.909 & 46.490 & 50.829 & 53.407 & 54.993 & 56.033 & 56.759 \end{array}$$ By how much does the standard molar enthalpy of $\mathrm{SO}_{2}(\mathrm{g})$ increase when the temperature is raised from $298.15 \mathrm{K}$ to $1500 \mathrm{K}$ ? Hint: Fit the data to an expression of the form of $C_{p, m}^{\ominus}(T)=a+b T+c / T^{2},$ note the values of the coefficients, then use the approach in Example 2 B. 2 to calculate the change in standard molar enthalpy.

The following data show how the standard molar constant-pressure heat capacity of sulfur dioxide varies with temperature:
$$\begin{array}{lllllll}
T / \mathrm{K} & 300 & 500 & 700 & 900 & 1100 & 1300 & 1500 \\
C_{p, m}^{\circ} /\left(\mathrm{JK}^{-1} \mathrm{mol}^{-1}\right) & 39.909 & 46.490 & 50.829 & 53.407 & 54.993 & 56.033 & 56.759
\end{array}$$
By how much does the standard molar enthalpy of $\mathrm{SO}_{2}(\mathrm{g})$ increase when the temperature is raised from $298.15 \mathrm{K}$ to $1500 \mathrm{K}$ ? Hint: Fit the data to an expression of the form of $C_{p, m}^{\ominus}(T)=a+b T+c / T^{2},$ note the values of the coefficients, then use the approach in Example 2 B. 2 to calculate the change in standard molar enthalpy.

Key Concepts

Standard Molar Enthalpy Change (?H°)

This concept refers to the energy change that occurs when one mole of a substance is heated at constant pressure over a temperature range. It is directly related to the heat capacity by an integral, which sums the small contributions of energy change at each infinitesimal temperature increase. Calculating ?H° is essential in thermodynamics as it allows for the prediction of energy requirements or releases during heating processes.

Heat Capacity at Constant Pressure (Cp,m°)

The standard molar heat capacity at constant pressure, Cp,m°, is the amount of heat required to raise the temperature of one mole of a substance by one degree at constant pressure. It provides insight into the substance’s ability to store thermal energy. In thermochemical calculations, Cp,m° is often a function of temperature, and understanding its behavior is crucial for accurately predicting changes in enthalpy over a temperature range.

Curve Fitting and Functional Approximation

In many thermodynamic analyses, experimental data are provided at discrete temperatures, requiring the use of curve fitting to obtain a continuous function. By fitting the Cp,m° data to an expression such as a + bT + c/T², one can derive coefficients that capture the temperature dependence accurately. This process simplifies subsequent integrations and allows for extrapolation or interpolation between data points.

Integration of the Heat Capacity Function

Once a functional form for the heat capacity is established, calculating the change in enthalpy over a temperature range involves integrating this function with respect to temperature. The integration of each term—constant, linear, and reciprocal-square—leads to analytical expressions that, when evaluated at the endpoints, yield the net change in enthalpy. This technique is a fundamental method in thermodynamics for relating microscopic properties to macroscopic energy changes.

Transcript

Please give Ace some feedback