If the heat capacity of a substance is correctly represented...

If the heat capacity of a substance is correctly represented by an equation of the form, $$ C_P=A+B T+C T^2 $$ show that the error resulting when $\left\langle C_P\right\rangle_H$ is assumed equal to $C_P$ evaluated at the arithmetic mean of the initial and final temperatures is $C\left(T_2-T_1\right)^2 / 12$.

If the heat capacity of a substance is correctly represented by an equation of the form,
$$
C_P=A+B T+C T^2
$$
show that the error resulting when $\left\langle C_P\right\rangle_H$ is assumed equal to $C_P$ evaluated at the arithmetic mean of the initial and final temperatures is $C\left(T_2-T_1\right)^2 / 12$.

Key Concepts

Heat Capacity at Constant Pressure

This concept refers to the amount of heat needed to raise the temperature of a substance by one degree while keeping the pressure constant. It is a fundamental property in thermodynamics that is used to compute changes in enthalpy. When the heat capacity varies with temperature—as represented by a polynomial expression—it must be integrated over the temperature range to accurately determine the energy change.

Average Value of a Function Over an Interval

The average value of a function over an interval is defined as the integral of the function divided by the length of the interval. In thermal calculations, this average is often used to approximate the effective heat capacity over a range of temperatures. However, for nonlinear functions, such as a quadratic temperature dependence, the arithmetic mean of the endpoints does not exactly represent the true average of the function over the interval.

Integration of Temperature-Dependent Functions

When the heat capacity is expressed as a function of temperature, calculating the total heat required to change the temperature involves integrating this function over the specified temperature interval. This process takes into account the variation of heat capacity with temperature, particularly when the dependency includes higher-order terms like T².

Error Analysis in Approximation Methods

Error analysis in this context involves quantifying the difference between an approximation and the exact value obtained from integration. For a quadratic temperature dependence in heat capacity, evaluating the function at the arithmetic mean of the initial and final temperatures leads to a systematic error. Deriving this error necessitates comparing the result of the integral to the value obtained by the midpoint approximation, and for the quadratic term, this error turns out to be proportional to the square of the temperature difference divided by 12.

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