SIMPLE GAS LAWS

 Introduction 

The gas laws that we are going to discuss are the results of experiments carried out by several scientists on the physical properties of gases. In these experiments, relations among the variables of pressure, temperature, volume and amount of gases are considered and the results provide valuable information on the gaseous state of matter in turn helping the mankind in many ways.

Ideal gas and Ideal gas equation

When it is assumed that intermolecular forces do not exist among the molecules of a gas,such a gas is called an ideal gas. That is molecules in an ideal gas do not exhibit attraction or repulsion among them. Furthermore, the volume of ideal gas particles is considered negligible when compared to the volume of the container.The absolute temperature (T), pressure (P), volume (V) and the amount (n, moles) of a gas are the factors that affect the behaviour of a gas. P, T, V and n are related by the expression

                       𝑷𝑽 = 𝒏𝑹𝑻

The value of the constant R for one mole of an ideal gas can be calculated under theconditions of 0 °C and 1 atm as given below: (At 0 °C and 1 atm, the volume of one mole of ideal gas is 22.414 dm3)

𝑅 =𝑃 𝑉/𝑛 𝑇

=101325 Pa × 22.414 × 10−3 m3/1 mol × 273.15 K

= 8.314 J K−1 mol−1

We can see that the ideal gas equation is a relationship among four variables and it describes the state of any gas. Therefore, it is also called equation of state.

The ideal gas law allows us to determine any one of the quantities volume, pressure, temperature, or moles of the gas when the other three are given. If the amount of moles of the gas is known, we can also calculate its mass using its molar mass. Further it can also be used to determine the density of a gas. It is important to keep in mind that all the other quantities must be in units that match the value used for the ideal gas constant. Usually pressure can be expressed in several units such as atm, Pa, bar, torr, etc.

 There are insights into the ideal gas equation that it can be expressed in different forms to estimate mass and density of a given gas with simple modifications as shown below. 

PV = n RT and we can write,

               P = (𝑛/𝑉)𝑅𝑇

                    𝑷 = 𝑪𝑹𝑻, 

where C is the concentration

We can also write PV = nRT as

               PV = (𝒎/𝑴)𝑹𝑻 ,

where, m is the mass and M is the molar mass of the gas.

Also we can write the above as;

               P = (1/M)(𝑚/𝑣) 𝑅𝑇

                   P = 𝒅 𝑹𝑻/𝑴

where, d is the density and 𝑑 =𝑚/𝑣

1.Boyle law👇

That is the “pressure of a fixed amount (mass) of gas at constant temperature inverselyvaries with (or proportional to) the volume of the gas”. This is known as the Boyle Law(1627-1691) which was named after Robert Boyle, an Irish scientist in the seventeenth century who studied change in volume of a gas when pressure of a gas is varied under constant temperature conditions. Mathematical form of it, is given below.

                     𝑃 ∝1/𝑉

            𝑜𝑟 𝑃 =𝑘/𝑉; k is a constant

The ideal gas law can be used to derive Boyle law as follows.

                         PV = nRT

If the amount of the gas and temperature of the system are kept constant, then the product nT is a constant. Since R is also a constant, then the product, nRT = k (a constant)

                        PV= k

It means that “at constant temperature, the product of pressure and volume of a fixed amount of gas is constant”. This is another way of expressing Boyle law.If a fixed amount of gas at constant temperature T occupying volume V1 at pressure P1undergoes change, so that volume becomes V2 and pressure becomes P2, then according to Boyle law             

                    P1V1 = P2V2

2.Charles law 

Investigations by the scientists Jacques Charles and Joseph Gay-Lussac have showed that for a fixed amount (mass) of a gas at constant pressure volume of a gas increases on heating and decreases on cooling. It was also found that for each degree change (rise or fall) in temperature, volume of a gas changes (increases or decreases) by a factor of 1/273.15 of the original volume of the gas at 0 C.Assume that volumes of the gas at 0 °C and at t °C are V0 and Vt respectively, then we can write,

𝑉𝑡 = 𝑉0 + (𝑡/273.15)𝑉0 = 𝑉0 (1 +𝑡/273.15) = 𝑉0 (273.15 + 𝑡/273.15 )

At this stage, a new scale of temperature is defined such that t °C on new scale is given by

              Tt = 273.15 + t

and 0 °C will be given by T0 = 273.15

This new temperature scale is called the Kelvin temperature scale (K) or absolute temperature scale. -273.15 °C (0 K) is also defined as the thermodynamic zero, which is the lowest theoretically reachable temperature.

By applying this temperature scale, we can rewrite the relation 

       𝑉𝑡 = 𝑉0 (273.15 + 𝑡/273.15 ) as, 

                     𝑉𝑡 = 𝑉0 (𝑇𝑡/𝑇0) 

Hence, 

                       𝑉𝑡/𝑉0=𝑇𝑡/𝑇0

For a general case when the change occurs from (V1, T1) to (V2, T2) at constant pressure

                       𝑉1/𝑉2=𝑇1/𝑇2

This can be rearranged as 

                𝑉1/𝑇1=𝑉2/𝑇2

𝑉/𝑇= constant or 𝑉 = 𝑘 𝑇

Therefore, “the volume of a fixed amount of gas under constant pressure is directly proportional to the absolute temperature of the gas.” This is called Charles law.

Further, the ideal gas law can be used to study the effect of temperature on the volume of a gas if the pressure of the system is kept constant for a fixed amount of a gas. The ideal gas law can be rearranged as follows;

                       PV = nRT

                       V = nRT/ P

When the pressure of a fixed mass of gas is constant, 𝑛𝑅/𝑃is constant.

                V ∝ T or 𝑽 = 𝒌𝑻

Let’s consider the equation 𝑉𝑡 = 𝑉0 (273.15+ 𝑡/273.15 ) and substitute t = -273.15, where we get the volume of the gas equal to zero meaning that the gas will not exist. Therefore, we can understand that all the gases get liquefied before this temperature is reached. The lowest hypothetical or imaginary temperature at which gases are supposed to occupy zero volume is called absolute zero.

3.Avagardo law

Upon the developments of Boyle and Charles laws, in 1811 Italian scientist Amedeo Avogadro tried to combine conclusions of those with the amount and volume of a gas andpostulated a new hypothesis which is now known as Avogadro law. It states that equalvolumes of all gases under the same conditions of temperature and pressure contain equal number of moles (Avogadro Law).

                      i.e. V ∝ n

        or we can write V = k n

The number of molecules in one mole of a gas has been determined to be 6.022 * 10^23and also known as Avogadro constant (denoted as NA or L).

Avogadro law can be easily understood with the help of the ideal gas law as follows.

                        𝑃𝑉 = 𝑛𝑅𝑇

                       𝑉 =(𝑅𝑇/𝑃)× 𝑛

         𝑉 =(𝑅𝑇/𝑃)×(𝑁/𝑁𝐴)=𝑅𝑇/𝑃𝑁𝐴× 𝑁

Here, N and NA are the number of molecules of the gas and the Avogadro constantrespectively. By applying the above relationships to equal volumes of gases P and Q at the same temperature and pressure,

                   𝑉𝑃 =𝑅𝑇/𝑃𝑁𝐴× 𝑁𝑃

                  𝑉𝑄 =𝑅𝑇/𝑃𝑁𝐴× 𝑁𝑄

At constant P and T, we can write (as R and NA are constants)Simply it says that for a gas at constant temperature and pressure equal volumes of gases have equal number of molecules. i.e. V∝N

It is useful to understand that the gas laws discussed above can also be used to obtainthe ideal gas equation for a given volume V of a gas. 

Boyel Law ∶ 𝑉 ∝1/𝑃− − − − − (1)

Charles Law ∶ 𝑉 ∝ 𝑇 − − − − − (2)

Avogardro Law ∶ 𝑉 ∝ 𝑛 − − − − − (3)

The only equation that fulfills (1), (2) and (3) is,

                   𝑉 ∝(𝑛/T)𝑃

                    𝑃𝑉/𝑛 𝑇= 𝑘

                   When 𝑘 = 𝑅

                      𝑃𝑉 = 𝑛𝑅𝑇

This video is taken from youtube from the channel of crash course

    To be continued.............

Made with the reference of chemistry resource book physical chemistry part -1 @www.nie.lk