An expression for the value of cp for carbon dioxide as...

An expression for the value of $c_{p}$ for carbon dioxide as function of temperature is \[c_{p}=286-\frac{1.15 \times 10^{5}}{T}+\frac{2.49 \times 10^{6}}{T^{2}}\] where $c_{p}$ is in $(\mathrm{ft} \cdot \mathrm{lb}) /\left(\mathrm{lbm} \cdot^{\circ} \mathrm{R}\right)$ and $T$ is in $^{\circ} \mathrm{R}$. Compare the change in enthalpy of carbon dioxide using the constant value of $c_{p}$ (see Table 1.7 ) with the change in cnthalpy of carbon dioxide using (a) $10^{\circ} \mathrm{R}$ the expression above, for $T_{2}-T_{1}$ equal to (b) $1000^{\circ} \mathrm{R}$ (c) $3000^{\circ} \mathrm{R}$. Set $T_{1}=540^{\circ} \mathrm{R}$

An expression for the value of $c_{p}$ for carbon dioxide as function of temperature is
\[c_{p}=286-\frac{1.15 \times 10^{5}}{T}+\frac{2.49 \times 10^{6}}{T^{2}}\]
where $c_{p}$ is in $(\mathrm{ft} \cdot \mathrm{lb}) /\left(\mathrm{lbm} \cdot^{\circ} \mathrm{R}\right)$ and $T$ is in $^{\circ} \mathrm{R}$. Compare the change in enthalpy of carbon dioxide using the constant value of $c_{p}$ (see Table 1.7 ) with the change in cnthalpy of carbon dioxide using
(a) $10^{\circ} \mathrm{R}$ the expression above, for $T_{2}-T_{1}$ equal to
(b) $1000^{\circ} \mathrm{R}$
(c) $3000^{\circ} \mathrm{R}$. Set $T_{1}=540^{\circ} \mathrm{R}$

Key Concepts

Specific Heat Capacity

Specific heat capacity is a thermodynamic property that quantifies the amount of heat required to change the temperature of a unit mass of a substance by one degree. It is a key parameter in energy balance and thermal calculations, and its value may vary with temperature, impacting the accuracy of energy change computations in thermodynamic processes.

Temperature Dependence of Specific Heat

When specific heat capacity varies with temperature, it is expressed as a function of temperature rather than a constant value. This reflects the changing ability of a material to store energy as its temperature varies, necessitating more complex calculations to accurately determine properties such as enthalpy changes over wide temperature ranges.

Enthalpy Change Calculation

Enthalpy change is calculated by integrating the specific heat capacity with respect to temperature. When cp is temperature-dependent, the integration must consider its variation over the temperature range of interest, contrasting with simpler calculations that use a constant cp which assumes a linear relationship between enthalpy change and temperature difference.

Integration of Temperature-Dependent Functions

The process of integrating a temperature-dependent specific heat expression is essential in thermodynamics for accurately capturing the non-linear relationship between heat input and temperature change. This integration accounts for the variable nature of cp with temperature and provides a more precise evaluation of the energy change in a system.

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