In a chemical experiment

MATHEMATICS · Multi-part · A Level · Mathematics · Differentiation · Differentiation

Problem

In a chemical experiment, the absolute temperature

T

(in kelvins) and pressure

P

(in atmospheres) are related by the implicit equation

T^{0.5P}=400\,e^{0.02P},\qquad T>0,\; P>0.

This relationship holds while the reaction proceeds in a sealed chamber. (a) Explain why

\dfrac{dT}{dP}

exists for all

(P,T)

satisfying the given equation with

T>0

and

P>0

. (b) Derive an expression for

\dfrac{dT}{dP}

in terms of

P

and

T

. (c) When the pressure is

P=10\text{ atm}

, determine the corresponding temperature

T

(accurate to three decimal places) and compute

\dfrac{dT}{dP}

at that pressure. Show all numerical work leading to your approximations. (d) Interpret the sign of

\dfrac{dT}{dP}

found in part (c) within the physical context of the experiment. What does it indicate about how the temperature changes as pressure increases near

P=10\text{ atm}