An Introduction to Gas Laws , Expansion and compression of gases
Introduction
Air, steam and other working substances used in thermodynamic systems pass through a cycle of processes as heat energy is changed into work and vice versa. Expansion and compression processes
often form an essential part of the cycle and we will now consider the laws, principles and properties associated with them.
One of the basic properties of a working substance is its temperature. Temperature is a measure of the hotness of a substance, which is directly proportional to the kinetic energy of its individual
molecules. Thermodynamic temperature is measured on the kelvin scale and is also known as absolute temperature. It has for its zero, the absolute zero of temperature at which all molecular movement ceases and the molecules of a substance have zero kinetic energy.
The SI unit of temperature is the kelvin, whose symbol is K. It is defined as the temperature interval between absolute zero and the triple point of water divided by 273.16. The triple point of water
occurs at a very low pressure, where the boiling point has been depressed to meet the freezing point. In this condition, ice, water and steam are able to exist together in the same container, which is
how the name arises. The kelvin is exactly the same temperature interval as the degree celsius (⁰C). The difference between the two scales lies in the points chosen for their definition. The degree
celsius is defined as the temperature interval between the freezing and boiling points of water at standard atmospheric pressure of 101 325 Pa, divided by 100. The freezing point of water at this
standard pressure is slightly below the triple point, being exactly 273K
The SI unit of pressure is the pascal (Pa). A pressure of 1 Pa is exerted when a force 1 N is evenly applied at right angles to a surface area of 1m². Another unit, which is widely used, is the ‘bar’. This is almost equal to standard atmospheric pressure.
1 bar=100 000 Pa or 1 × 10⁵ Pa or 100 kPa
Pressure-measuring devices such as mechanical gauges and the manometers measure the difference between the pressure inside a container and the outside atmospheric pressure. This, as you will recall, is known as gauge pressure.
In gas calculations, however, it is often the total or absolute pressure, which must be used. This is obtained by adding atmospheric pressure to the recorded gauge pressure (figure 1), that is :
absolute pressure=gauge pressure+atmospheric pressure
Reference has been made above to standard atmospheric pressure. A substance is said to be at standard temperature and pressure (STP) when its temperature is 0⁰C or 273 K, and its pressure is
101 325 Pa. This definition has international acceptance.
Sometimes a substance is said to be at normal temperature and pressure. A substance is at normal temperature and pressure (NTP) when its temperature is 15⁰C or 288 K, and its pressure is
again 101 325 Pa. This definition is used in the United Kingdom and other countries with a temperate climate.
the density of a working substance is often required for thermodynamic calculations (figure 2) . density is the mass per unit volume of a substance whose unit is kgm⁻³.
Sometimes however, and particularly in steam calculations, it is more convenient to use specific volume. This is the volume per unit mass of a substance whose unit is m³ kg⁻1. Specific volume, vs is thus the reciprocal of density, that is
specific volume=volume occupied by 1 kg of a substance
For a substance of mass m kg and volume V m³, its specific volume will be given by
The gas laws
The gas laws which we need to consider are Boyle’s law, Charles’ law , Avogadro’s law, which is also called Avogadro’s hypothesis. Combined and ideal gas laws , and Gay-Lussac's law
Boyle’s law
This states that the volume of a fixed mass of gas is inversely proportional to its absolute pressure provided that its temperature is constant. When plotted on a graph of absolute pressure against
volume, the process appears as shown in Figure 4. Expansion and compression processes which take place at constant temperature, according to Boyle’s law, are known as isothermal processes.
For any two points on the curve whose co-ordinates are p1V1 and p2V2, it is found that
This states that the volume of a fixed mass of gas is proportional to its absolute temperature provided that its pressure is constant. When plotted on a graph of absolute temperature against volume, the
process appears as shown in Figure 5.
Expansion and compression processes which take place according to Charles’ law are referred to as constant pressure or isobaric processes. For any two points on the curve whose co-ordinates are
T1V1 and T2V2, it is found that
You will note from Figure 5 that in theory, the volume of the gas should decrease uniformly until at absolute zero, which is the origin of the graph, its volume would also be zero. This is how
an ideal gas would behave.
Real gases obey the gas laws fairly closely at the temperatures and pressures normally encountered in
power and process plant but at very low temperatures they liquefy, and may also solidify, before reaching absolute zero.
The general gas equation
This equation may be applied to any fixed mass of gas which undergoes a thermodynamic process taking it from initial conditions p1, V1 and T1 to final conditions p2,V2 and T2. Suppose the
gas expands first according to Boyle’s law to some intermediate volume V. Let it then expand further according to Charles’ law to its final volume V2.
Figure 6 shows the processes plotted on a graph of absolute pressure against volume. For the initial expansion according to Boyle’s law,
p1 V1=p2 V
For the final stage according to Charles’ law,
V/ T1 = V2 / T2
⇒
V = (V2 T1) / T2
⇒ p1 V1 = p2 (V2 T1) / T2
Substituting in equation
This is the general gas equation which can be used to relate any two sets of conditions for a fixed mass of gas, irrespective of the process or processes which have caused the change. The constant in equation is the product of two quantities. One of them is the mass m kg of the gas. The other is a constant for the particular gas known as its characteristic gas constant.
The characteristic gas constant R has unit of joules per kilogram kelvin (J kg⁻¹K⁻¹) and is related to the molecular weight of the gas. Equation can thus be written as
In this form it is known as the characteristic gas equation which is particularly useful for finding the mass of a gas whose volume, absolute pressure and absolute temperature are known.
The general gas equation can be used to relate any two sets of conditions for a fixed mass of gas, irrespective of the process or processes which have taken place.
Avogadro’s hypothesis
This states that equal volumes of different gases at the same temperature and pressure contain the same number of molecules. It is called a hypothesis because it cannot be directly proved. It is impossible to count the very large number of molecules, even in a small volume, but there is lots of evidence to indicate that the assumption is true. Suppose we have three vessels of equal volume
which contain three different gases at the same temperature and pressure, as shown in Figure 7.
Avogadro's law states that the volume occupied by an ideal gas is directly proportional to the number of molecules of the gas present in the container. This gives rise to the molar volume of a gas, which at STP (273.15 K, 1 atm) is about 22.4 L. The relation is given by
Gay-Lussac's law
Dalton's law
Some specific values of n correspond to particular cases:
∎ n=0 is for an isobaric process,
∎ n=+∞ is for an isochoric process.
In addition, when the ideal gas law applies:
∎ n=1 is for an isothermal process,
∎ n= 𝛾 is for an isentropic process.
Where 𝛾 is the ratio of the heat capacity at constant pressure Cp to heat capacity at constant volume Cv
Introduction
Air, steam and other working substances used in thermodynamic systems pass through a cycle of processes as heat energy is changed into work and vice versa. Expansion and compression processes
often form an essential part of the cycle and we will now consider the laws, principles and properties associated with them.
molecules. Thermodynamic temperature is measured on the kelvin scale and is also known as absolute temperature. It has for its zero, the absolute zero of temperature at which all molecular movement ceases and the molecules of a substance have zero kinetic energy.
The SI unit of temperature is the kelvin, whose symbol is K. It is defined as the temperature interval between absolute zero and the triple point of water divided by 273.16. The triple point of water
occurs at a very low pressure, where the boiling point has been depressed to meet the freezing point. In this condition, ice, water and steam are able to exist together in the same container, which is
how the name arises. The kelvin is exactly the same temperature interval as the degree celsius (⁰C). The difference between the two scales lies in the points chosen for their definition. The degree
celsius is defined as the temperature interval between the freezing and boiling points of water at standard atmospheric pressure of 101 325 Pa, divided by 100. The freezing point of water at this
standard pressure is slightly below the triple point, being exactly 273K
The SI unit of pressure is the pascal (Pa). A pressure of 1 Pa is exerted when a force 1 N is evenly applied at right angles to a surface area of 1m². Another unit, which is widely used, is the ‘bar’. This is almost equal to standard atmospheric pressure.
1 bar=100 000 Pa or 1 × 10⁵ Pa or 100 kPa
Pressure-measuring devices such as mechanical gauges and the manometers measure the difference between the pressure inside a container and the outside atmospheric pressure. This, as you will recall, is known as gauge pressure.
In gas calculations, however, it is often the total or absolute pressure, which must be used. This is obtained by adding atmospheric pressure to the recorded gauge pressure (figure 1), that is :
 |
| FIG 1 |
absolute pressure=gauge pressure+atmospheric pressure
Reference has been made above to standard atmospheric pressure. A substance is said to be at standard temperature and pressure (STP) when its temperature is 0⁰C or 273 K, and its pressure is
101 325 Pa. This definition has international acceptance.
Sometimes a substance is said to be at normal temperature and pressure. A substance is at normal temperature and pressure (NTP) when its temperature is 15⁰C or 288 K, and its pressure is
again 101 325 Pa. This definition is used in the United Kingdom and other countries with a temperate climate.
the density of a working substance is often required for thermodynamic calculations (figure 2) . density is the mass per unit volume of a substance whose unit is kgm⁻³.
 |
| FIG 2 |
Sometimes however, and particularly in steam calculations, it is more convenient to use specific volume. This is the volume per unit mass of a substance whose unit is m³ kg⁻1. Specific volume, vs is thus the reciprocal of density, that is
specific volume=volume occupied by 1 kg of a substance
For a substance of mass m kg and volume V m³, its specific volume will be given by
The gas laws which we need to consider are Boyle’s law, Charles’ law , Avogadro’s law, which is also called Avogadro’s hypothesis. Combined and ideal gas laws , and Gay-Lussac's law
 |
| FIG 3 |
Boyle’s law
This states that the volume of a fixed mass of gas is inversely proportional to its absolute pressure provided that its temperature is constant. When plotted on a graph of absolute pressure against
volume, the process appears as shown in Figure 4. Expansion and compression processes which take place at constant temperature, according to Boyle’s law, are known as isothermal processes.
 |
| FIG 4 |
p×V = constant
Charles’ lawThis states that the volume of a fixed mass of gas is proportional to its absolute temperature provided that its pressure is constant. When plotted on a graph of absolute temperature against volume, the
process appears as shown in Figure 5.
 |
| FIG 5 |
T1V1 and T2V2, it is found that
an ideal gas would behave.
Real gases obey the gas laws fairly closely at the temperatures and pressures normally encountered in
power and process plant but at very low temperatures they liquefy, and may also solidify, before reaching absolute zero.
The general gas equation
This equation may be applied to any fixed mass of gas which undergoes a thermodynamic process taking it from initial conditions p1, V1 and T1 to final conditions p2,V2 and T2. Suppose the
gas expands first according to Boyle’s law to some intermediate volume V. Let it then expand further according to Charles’ law to its final volume V2.
 |
| FIG 6 |
p1 V1=p2 V
For the final stage according to Charles’ law,
V/ T1 = V2 / T2
⇒
V = (V2 T1) / T2
⇒ p1 V1 = p2 (V2 T1) / T2
Substituting in equation
This is the general gas equation which can be used to relate any two sets of conditions for a fixed mass of gas, irrespective of the process or processes which have caused the change. The constant in equation is the product of two quantities. One of them is the mass m kg of the gas. The other is a constant for the particular gas known as its characteristic gas constant.
The characteristic gas constant R has unit of joules per kilogram kelvin (J kg⁻¹K⁻¹) and is related to the molecular weight of the gas. Equation can thus be written as
In this form it is known as the characteristic gas equation which is particularly useful for finding the mass of a gas whose volume, absolute pressure and absolute temperature are known.
The general gas equation can be used to relate any two sets of conditions for a fixed mass of gas, irrespective of the process or processes which have taken place.
Avogadro’s hypothesis
This states that equal volumes of different gases at the same temperature and pressure contain the same number of molecules. It is called a hypothesis because it cannot be directly proved. It is impossible to count the very large number of molecules, even in a small volume, but there is lots of evidence to indicate that the assumption is true. Suppose we have three vessels of equal volume
which contain three different gases at the same temperature and pressure, as shown in Figure 7.
 |
| FIG 7 |
Avogadro’s hypothesis states that each vessel contains the same number of molecules. Let this number be N. The mass of gas in each vessel will be different, and given by
mass of gas in vessel = number of molecules × molecular mass
m=N × M
Applying the characteristic gas equation to the first vessel gives
Substituting for m from equation for all gases in three vessels
Equating gives
The actual mass of a molecule is very small and so in place of it, engineers and chemists use the kilogram-molecule or kmol. This is the mass of the substance, measured in kilograms, which is
numerically equal to its molecular weight. The molecular weight is the weight or mass of a molecule of the gas relative to the weight or mass of a single hydrogen atom. Typical values for some common
gases are shown
The product MR is found to have a value of 8314 J kmol⁻¹ and is called the universal gas constant. It can be used to calculate the value of the characteristic gas constant R for a particular gas of known molecular weight.
M R = 8314
e.g. the molecular weight of oxygen is 32 and so the value of its characteristic gas constant will be
R= 8314 / 32 =260 J kg⁻¹ K⁻¹
 |
| FIG 8 |
A kilogram-molecule or kmol of a substance is the mass in kilograms which is numerically equal to its bmolecular weight.
Gay-Lussac's law, Amontons' law or the pressure law for a given mass and constant volume of an ideal gas, the pressure exerted on the sides of its container is directly proportional to its absolute temperature.
As a mathematical equation, Gay-Lussac's law is written as either:
In chemistry and physics, Dalton's law (also called Dalton's law of partial pressures) states that in a mixture of non-reacting gases, the total pressure exerted is equal to the sum of the partial pressures of the individual gases ( figure 9).
 |
| FIG 9 |
The Total Gas Pressure shown in Container T is determined by adding the Partial Pressure of Gases A, B, and C.
Mathematically, the pressure of a mixture of non-reactive gases can be defined as
Polytropic expansion
A fixed mass of gas can expand from an initial pressure p1 and volume V1 in an infinite number of ways. This is what mean by the word polytropic. Figure 10 shows some of possible expansion processes.
 |
| FIG 10 |
The particular expansion curve followed by a gas depends on the amount and direction of the heat transfer that takes place during the expansion process. All curves have an equation of the general form
For most practical expansion and compression processes, the value of the index n ranges from low negative values to positive values of around 1.5 . The values of n indicated as :
Some specific values of n correspond to particular cases:
∎ n=0 is for an isobaric process,
∎ n=+∞ is for an isochoric process.
In addition, when the ideal gas law applies:
∎ n=1 is for an isothermal process,
∎ n= 𝛾 is for an isentropic process.
Where 𝛾 is the ratio of the heat capacity at constant pressure Cp to heat capacity at constant volume Cv
The value of the index n in a polypropic process depends on the magnitude and direction of the heat transfer that takes place.
thanks for your visit