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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

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BuyFindarrow_forward

Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621
Textbook Problem

Boyle’s Law states that when a sample of gas is compressed at a constant temperature, the product of the pressure and the volume remains constant: P V = C

(a) Find the rate of change of volume with respect to pressure.

(b) A sample of gas is in a container at low pressure and is steadily compressed at constant temperature for 10 minutes. Is the volume decreasing more rapidly at the beginning or the end of the 10 minutes? Explain.

(c) Prove that the isothermal compressibility (see Example 5) is given by β = 1 / P .

To determine

Part (a)

To find: the rate of change of volume with respect to pressure.

Explanation

1) Formula:

Quotient rule,  ddxuv=u'v-uv'v2

2) Given:

PV=C

3) Calculation:

Solve for V from given equation,

V=CP

Now take derivative with respect to P,

d</

To determine

Part (b)

To check: Is the volume decreasing more rapidly at the beginning or at the end of 10 minutes

To determine

Part (C)

To prove: that the isothermal compressibility is given by Β=1P

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