At one atmosphere, the specific heat capacity of water at...

At one atmosphere, the specific heat capacity of water at constant pressure is given to four significant figures by the formula $$ \begin{aligned} \frac{c_p(\theta)}{c}= & 0.996185+0.0002874\left(1+\frac{\theta}{100}\right)^{5.26} \\ & +0.011160 e^{-0.0829 \theta} \end{aligned} $$ where $c=4185.5 \mathrm{JK}^{-1} \mathrm{~kg}^{-1}$ and $\theta$ is the temperature in degrees Celsius. Find the heat energy required to raise the temperature of 50 grams of water from $0^{\circ} \mathrm{C}$ to $100^{\circ} \mathrm{C}$.

At one atmosphere, the specific heat capacity of water at constant pressure is given to four significant figures by the formula
$$
\begin{aligned}
\frac{c_p(\theta)}{c}= & 0.996185+0.0002874\left(1+\frac{\theta}{100}\right)^{5.26} \\
& +0.011160 e^{-0.0829 \theta}
\end{aligned}
$$
where $c=4185.5 \mathrm{JK}^{-1} \mathrm{~kg}^{-1}$ and $\theta$ is the temperature in degrees Celsius. Find the heat energy required to raise the temperature of 50 grams of water from $0^{\circ} \mathrm{C}$ to $100^{\circ} \mathrm{C}$.

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Key Concepts

Calculation of Energy Transfer

The total heat energy needed to raise the temperature of a given mass of a substance is calculated by multiplying the mass by the integral of the temperature-dependent specific heat capacity over the range of temperatures. This method ensures that the variation in energy required at different temperatures is properly accounted for.

Integration in Thermodynamics

When a thermodynamic property such as specific heat capacity varies with temperature, determining the total energy required for a temperature change involves integrating the specific heat function over the temperature interval. This integral sums the incremental energy contributions across the changing temperature.

Specific Heat Capacity

Specific heat capacity is the amount of energy required to raise the temperature of a unit mass of a substance by one degree. It is a fundamental thermodynamic property that helps in understanding how different materials absorb and store energy.

Temperature Dependence of Material Properties

In many situations, the specific heat capacity is not constant; rather, it varies with temperature due to the changing dynamics at the molecular level. This temperature dependence must be taken into account for accurate calculations of energy transfer over a range of temperatures.

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