A fixed quantity of an ideal gas is taken from state to state along two different…

PHYSICS · Multi-part · A Level · Physics · Further mechanics and thermal physics · Further mechanics and thermal physics

Problem

A fixed quantity of an ideal gas is taken from state

A

to state

B

along two different paths described below. In both cases the initial state has pressure

P_{1}

and volume

V_{1}

, and the final state has pressure

P_{2}

and volume

V_{2}

where

P_{2}=\tfrac12P_{1}

and

V_{2}=2V_{1}

. (i) Path I is a direct isothermal expansion from

A

B

. Path II consists of an isobaric expansion from

(P_{1},V_{1})

(P_{1},2V_{1})

followed by an isochoric cooling to

(P_{2},2V_{1})

. Draw the two paths on the same

P

–

V

diagram and label them clearly. (ii) Show that Path I satisfies the mathematical form

PV=\text{constant}

and hence justify that it is isothermal. (iii) Using the ideal–gas equation, determine the temperature at state

B

for each path in terms of the initial temperature

T_{1}

. (iv) Explain, without using numerical integration, which of the two paths involves greater work done by the gas and give a qualitative reason based on the shapes of the curves on the

P

–

V

plane.

Related theory

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