A chemical reaction inside a sealed chamber releases heat

MATHEMATICS · Multi-part · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation

Problem

A chemical reaction inside a sealed chamber releases heat. The total heat released, measured in joules, is represented by the function

H(T)

, where

T

is the temperature inside the chamber measured in degrees Celsius. The temperature itself changes over time

t

(in seconds) according to a differentiable function

T(t)

. At a certain instant

t=t_{0}

, the following information is recorded:

\qquad H'(T(t_{0})) = 1.80\times10^{3}\,\text{J}/^{\circ}\text{C}

\qquad \displaystyle \frac{dT}{dt}\Big|_{t=t_{0}} = 0.040\,^{\circ}\text{C}/\text{s}

Investigators need to determine how quickly heat is being released with respect to time at this instant. (a) Derive an expression for

\displaystyle \frac{dH}{dt}

t=t_{0}

in terms of

H'(T(t_{0}))

and

\dfrac{dT}{dt}\Big|_{t=t_{0}}

, clearly indicating the units at each stage of your derivation. (b) Compute the numerical value of

\dfrac{dH}{dt}

t=t_{0}

, expressing your answer in watts (joules per second). Show your unit analysis in detail to verify that the final units simplify correctly to J/s. (c) Suppose temperature were instead recorded in kelvins, with

\theta = T + 273.15

. Explain whether

H'(\theta)

would have the same numerical value as

H'(T)

and justify your conclusion through unit analysis. (d) Write a concise statement interpreting the meaning of the value obtained in part (b) for the rate of heat release in the context of the experiment.

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