   Chapter 10, Problem 64AP

Chapter
Section
Textbook Problem

An expandable cylinder has its top connected to a spring with force constant 2.00 × 103 N/m (Fig. P10.64). The cylinder is Filled with 5.00 L of gas with the spring relaxed at a pressure of 1.00 atm and a temperature of 20.0°C. (a) If the lid has a cross-sectional area of 0.010 0 m2 and negligible mass, how high will the lid rise when the temperature is raised to 250°C? (b) What is the pressure of the gas at 250°C? Figure P10.64

(a)

To determine
The height up to which the lid rises.

Explanation

Given info:

The initial pressure is 1.0 atm.

The initial volume is 5.0 L.

The initial temperature is 20οC .

The final temperature is 250οC .

The force constant is 2×103N/m .

The cross sectional area is 0.010m2 .

Explanation:

From Ideal gas law,

(P2P1)(V2V1)=T2T1 (I)

• P1 and P2 are the initial and final pressures.
• T1 and T2 are the initial and final temperatures.
• V1 and V2 are the initial and final volumes.

Formula to calculate the final pressure is,

P2=P1+(khA) (II)

• k is the force constant.
• h is the height up to which the lid rises.
• A is the cross sectional area.

Formula to calculate the final volume is,

V2=V1+Ah (III)

Substitute Equations (II) and (III) in (I)

(1+khAP1)(1+AhV1)=T2T1

On Re-arranging the above equation,

(kP1V1)h2+(kAP1+AV1)h+(1T2T1)=0

Using the quadratic formula to solve for h,

h=(kAP1+AV1)±(kAP1+AV1)24(kP1V1)(1T2T1)2(kP1V1)

Substitute 1

(b)

To determine
The pressure of the gas at 250οC .

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