# Find the rate of change of volume with respect to pressure.

### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

#### Solutions

Chapter 3.8, Problem 21E

a.

To determine

## Find the rate of change of volume with respect to pressure.

Expert Solution

dVdP=CP2

### Explanation of Solution

Given information:

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: PV=C .

Calculation:

The equation that models the volume can be wriiten as V=CP . Use the power rule to find

dVdP:dVdP=CP2

b.

To determine

### Is the volume decreasing more rapidly at the beginning or the end of the 10 minutes?

Expert Solution

The volume is decreasing most rapidly at the beginning, when P2 is the smallest.

### Explanation of Solution

Given information:

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: PV=C .

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.

Calculation:

The equation that we just found for change volume indicates that as the pressure gets larger constant temperature, the rate at which the volume changes deceases.

Hence, the volume is decreasing most rapidly at the beginning, when P2 is the smallest.

c.

To determine

Expert Solution

β=1P Proved.

### Explanation of Solution

Given information:

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: PV=C .

Calculation:

Isothermal compressibility is defined as β=1VdVdP .Put this in our equation for dVdP:

β=1V(CP2)=CVP2

We know that PV=C , hence, put in this information:

β=PVP2Vβ=1P

Hence, proved.

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