Chapter 1, Problem 1.36E

Interpretation Introduction

(a)

Interpretation:

The value of the compressibility factor for an ideal gas is to be stated.

Concept introduction:

The ideal gas law considered the molecules of a gas as point particles with perfectly elastic collisions among them in nature. This works importantly well for gases at dilution and at low pressure in many experimental calculations. But the gas molecules are not performing as point masses, and there are situations where the properties of the gas molecules have measurable effect by experiments. Thus, a modification of the ideal gas equation was coined by Johannes D. van der Waals in $1873$ to consider size of molecules and the interaction forces among them. It is generally denoted as the van der Waals equation of state.

Answer to Problem 1.36E

The ideal gas equation is $Z=\frac{PV}{RT}$ and the compressibility factor are ‘one’.

Explanation of Solution

In many circumstances such as at low temperature and high pressures the gases deviate from the ideal gas equation. $PV=nRT$ They are considered as non-ideal or non-real gases. The behavior of non-ideal gases using more complicated equations of state. For one mole of ideal gas, the ideal gas equation can be written as,

$\frac{P\overline{V}}{RT}=1\cdot \cdot \cdot \text{(1)}$

The equation (1) can be written for non-ideality with correction as,

$P\overline{V}=ZRT\cdot \cdot \cdot \text{(2)}$

Where,

P = Pressure

$\overline{V}$= Molar volume

Z = Compressibility factor

R = Universal gas constant

T = Temperature

Therefore, the compressibility factor can be written as,

$Z=\frac{P\overline{V}}{RT}\cdot \cdot \cdot \text{(3)}$

This is simplest form of equation of state of real gas. The key factor of equation (3) is that the compressibility factor, Z, is not a constant. Basically, the value of ‘Z’ varies from one gas to another gas as well as varies with the pressure and temperature of the gas under consideration. Thus, it should be evaluated experimentally. The plot of ‘Z’ versus pressure at constant temperature of plot of ‘Z’ versus pressure at varying temperatures gives the readily obtaining interpolated values of ‘Z’ between the experimentally determined values.

The compressibility factor ‘Z’ can be expressed in another form as,

$Z=\frac{V_{\text{actual}}}{V_{\text{ideal}}}$

The factors affecting the compressibility values are;

1. When the gas pressure approaches 0, the value of Z tends toward 1. In this case all gases show ideal behavior.

2. When the gas pressure is at intermediate level, the value of Z is less than 1. In this case actual volumes to be less than the ideal values due to intermolecular forces of attraction.

3. When the gas pressure is high, the value of Z is greater than 1 and tends toward infinity. In this case the actual volumes to be greater than the ideal values due to intermolecular repulsive forces.

Conclusion

Thus, the value of the compressibility factor for an ideal gas is stated.

Interpretation Introduction

(b)

Interpretation:

‘The value varies with $p,V,T\text{or}n$the statement is to be verified.

Concept introduction:

The ideal gas law considered the molecules of a gas as point particles with perfectly elastic collisions among them in nature. This works importantly well for gases at dilution and at low pressure in many experimental calculations. But the gas molecules are not performing as point masses, and there are situations where the properties of the gas molecules have measurable effect by experiments. Thus, a modification of the ideal gas equation was coined by Johannes D. van der Waals in 1873 to consider size of molecules and the interaction forces among them. It is generally denoted as the van der Waals equation of state.

Answer to Problem 1.36E

Since, the compressibility factor Z is having the variables p, V, and T, its value will certainly vary with the terms of p, V, and T. Generally, the farther the value of Z is from ‘one’ the gas behaves less ideally.

Explanation of Solution

In many circumstances such as at low temperature and high pressures the gases deviate from the ideal gas equation $PV=nRT$. They are considered as non-ideal or non-real gases. The behavior of non-ideal gases using more complicated equations of state. For one mole of ideal gas, the ideal gas equation can be written as,

$\frac{P\overline{V}}{RT}=1\cdot \cdot \cdot \text{(1)}$

The equation (1) can be written for non-ideality with correction as,

$P\overline{V}=ZRT\cdot \cdot \cdot \text{(2)}$

Where,

P = Pressure

$\overline{V}$= Molar volume

Z = Compressibility factor

R = Universal gas constant

T = Temperature

Therefore, the compressibility factor can be written as,

$Z=\frac{P\overline{V}}{RT}\cdot \cdot \cdot \text{(3)}$

This is simplest form of equation of state of real gas. The key factor of equation (3) is that the compressibility factor, Z, is not a constant. Basically, the value of ‘Z’ varies from one gas to another gas as well as varies with the pressure and temperature of the gas under consideration. Thus, it should be evaluated experimentally. The plot of ‘Z’ versus pressure at constant temperature of plot of ‘Z’ versus pressure at varying temperatures gives the readily obtaining interpolated values of ‘Z’ between the experimentally determined values.

The compressibility factor ‘Z’ can be expressed in another form as,

$Z=\frac{V_{\text{actual}}}{V_{\text{ideal}}}$

The factors affecting the compressibility values are;

1. When the gas pressure approaches 0, the value of Z tends toward 1. In this case all gases show ideal behavior.

2. When the gas pressure is at intermediate level, the value of Z is less than 1. In this case actual volumes to be less than the ideal values due to intermolecular forces of attraction.

3. When the gas pressure is high, the value of Z is greater than 1 and tends toward infinity. In this case the actual volumes to be greater than the ideal values due to intermolecular repulsive forces.

Conclusion

Thus, ‘The value varies with $p,V,T\text{or}n$- the statement is verified.