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An introduction to the thermodynamics of ideal gases

Definitions Working substance (WS) = the WS is used as the carrier for heat energy. The state of the

WS is defined by the values of its properties, e.g. pressure, volume, temperature, internalenergy, enthalpy. These properties are also sometimes called functions of state.

Key facts An ideal gas is a WS which obeys Boyle's law, Charles' law, Amontons' law, Avogadro's

law, Joule's law of internal energy, Dalton's law of partial pressures, and has a constant

specific heat.Boyle's law states that if , then:

Charles' law states that if , then:

Amontons' law states that if , then:

The combined gas law affirms that:

The ideal gas law can be written for moles of gas as:

where is the universal gas constant.Avogadro's law states that equal volumes of gas contain, at the same temperature and

 pressure, the same number of molecules.Joule's law of internal energy states that the internal energy of an ideal gas is independent

of its pressure and volume, and depends only on its temperature.Dalton's law of partial pressures affirms that the total pressure exerted by a gaseous

mixture is equal to the sum of the partial pressures of each individual component of that

mixture.

The heat capacity at constant pressure is related to the heat capacity at constant

volume by:

where is the universal gas constant.

The heat capacity ratio is defined as:

The change in internal energy of moles of an ideal gas undergoing a change in

temperature of is given by:

The change in enthalpy of moles of an ideal gas undergoing a change in temperature of 

is given by:

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Constants 

An ideal gas is a working substance (WS) which obeys Boyle's law, Charles' law,Amontons' law, Avogadro's law, Joule's law of internal energy, Dalton's law of partial

 pressures, and has a constant specific heat. In order to obey all these laws, the WS would

not be able to change its state even at absolute zero. Therefore, the molecules of the WS 

would need to be so far apart that there are no intermolecular forces and no collisions.

At normal temperatures and pressures, the permanent gases (e.g. hydrogen, oxygen,

nitrogen) closely obey these laws. Therefore, these gases are called semi-perfect gases.

Ideal Gas Laws

 Boyle's law (also sometimes called Boyle-Mariotte's law) states that if the temperature of 

a fixed mass of gas is kept constant, then its pressure is inversely proportional to itsvolume. Expressed in mathematical terms, Boyle's law affirms that if is constant, then:

(1)

Boyle's law is illustrated in Figure 1 (to see the animation click on the thumbnail), whichshows the inverse proportionality between pressure and volume when mass and

temperature are kept constant.

Figure 1 (http://www.grc.nasa.gov)

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Charles'  law (also sometimes called Charles and Gay-Lussac's law) states that if the

 pressure of a fixed mass of gas is kept constant, then its volume is directly proportional to

its temperature. Translated into mathematical terms, this means that if is constant, then:

(2)Charles' law is depicted in Figure 2 (to see the animation click on the thumbnail), which

highlights the direct proportionality between volume and temperature when mass and pressure are kept constant.

Figure 2 (http://www.grc.nasa.gov)

 Amontons'  law (also known as the pressure-temperature law) states that if the volume of a

fixed mass of gas is kept constant, then its pressure is directly proportional to itstemperature. Expressed in mathematical terms, this law affirms that if is constant, then:

(3)

Boyle's law, Charles' law, and Amontons' law can be associated to devise the so-calledcombined gas law, which states that the ratio between the pressure-volume product andthe temperature of a fixed mass of gas remains constant, or, in mathematical terms, that:

(4)

The combined gas law can be expressed more generally for moles of gas as:

(5)or, furthermore, as:

(6)

where is a constant called the universal gas constant. Equation (6) is named the ideal 

 gas law, and represents the equation of state of an ideal gas. The value of in imperialis:

(7)

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while in metric it is:

(8)

We can also define the specific gas constant of a gas as the ratio between the universal

gas constant and the molar mass of that gas:

(9)

For example, the specific gas constant of dry air is:

(10)or, expressed in alternative units:

(11)

(12)

(13)Another characteristic law of ideal gases is Avogadro's law, which states that equal

volumes of gas contain, at the same temperature and pressure, the same number of 

molecules. Avogadro's law can be expressed in mathematical terms as:

(14)

where is the volume of the gas, and the number of moles of the gas. Joule's law of internal energy states that the internal energy of an ideal gas is independent

of its pressure and volume, and depends only on its temperature. In mathematical termsthis means that the internal energy is a function of the absolute temperature :

(15)

Yet another law characteristic of ideal gases is Dalton's law of partial pressures, which

states that the total pressure exerted by a gaseous mixture is equal to the sum of the

 partial pressure of each individual component of that mixture. This can be written inmathematical terms as:

(16)

where is the total pressure, is the partial pressure of component , and is the total

number of components of the gaseous mixture.

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Specific Heat Of Ideal Gases

The (molar) specific heat is the quantity of heat required to raise the temperature of one

mole of substance by one degree.In the case of gases, the specific heat depends on the way in which the gas is heated. For 

example, if it is allowed to do work, then the specific heat must be greater. We canimagine therefore an infinite number of specific heats. However, we will consider only

two of them: the specific heat at constant volume, and the specific heat at constant pressure.

Imagine a heating process at constant volume. The first law of thermodynamics states

that the change in internal energy of a system is equal to the heat added to the systemminus the work done by the system:

(17)In our case, the work done by the system is given by:

(18)while the heat added to the WS during a process at constant volume (the specific heat at

constant volume) can be written as:

(19)

where is the number of moles of the WS , the heat capacity at constant volume, andthe change in temperature.

By using (18) and (19) in equation (17), we obtain that:

(20)

However, we previously saw that the internal energy of an ideal gas is independent of its

 pressure and volume, and depends only on its temperature (see Joule's law of internalenergy, equation 15). Therefore, equation (20) becomes:

(21)or, by considering the datum as absolute zero:

(22)

 Now imagine a heating process at constant pressure. The heat added to the WS in this

case (the specific heat at constant pressure) is given by:

(23)

where is the heat capacity at constant pressure.Taking into account that the ideal gas law (6) can also be written as:

(24)

and also considering equations (21) and (23), the first law of thermodynamics (17)

 becomes:

(25)

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from which we obtain that:

(26)

or, furthermore, that:

(27)Hence, the universal gas constant relates the heat capacity at constant volume to the

heat capacity at constant pressure .

Also, let the ratio between and be denoted by :

(28)

where is called the heat capacity ratio.The value of varies depending on the degrees of freedom of the gas, which in turn is

related to its atomic composition. For example, monoatomic gases can rotate only about

their own axis, while diatomic gases can rotate about their own axis, as well as the two

atoms of the molecule about each other. Hence, for monoatomic gases (which have onlyone degree of rotational freedom) is approximately , for diatomic gases (which

have two degrees of rotational freedom) is , while for polyatomic gases is

approximately .

Enthalpy Of Ideal Gases

We know that the change in enthalpy of a system can be written as:

(29)

Taking into account equations (21) and (24), the change in enthalpy of an ideal gas

 becomes:

(30)

or, furthermore:

(31)

As (see equation 27), equation (31) becomes:

(32)

By considering the datum as absolute zero, we obtain the enthalpy of an ideal gas as: