`color{blue} ✍️`Properties of gases are easier to understand than those of solids and liquids. This is mainly because in a gas, molecules are far from each other and their mutual interactions are negligible except when two molecules collide.
`color{blue} ✍️`Gases at low pressures and high temperatures much above that at which they liquefy (or solidify) approximately satisfy a simple relation between their pressure, temperature and volume given by (see Ch. 11) for a given sample of the gas.
`color{blue} {PV = KT}`
........... (13.1)
`color{blue} ✍️`Here `T` is the temperature in kelvin or (absolute) scale. K is a constant for the given sample but varies with the volume of the gas.
`color{blue} ✍️`If we now bring in the idea of atoms or molecules then K is proportional to the number of molecules, (say) N in the sample. We can write `K = N k` . Observation tells us that this k is same for all gases. It is called Boltzmann constant and is denoted by `k_B`.
As
`color {blue} {(P_1V_1)/(N_1T_1) = (P_2V_2)/(N_2T_2) =} " constant "color{blue} { = k_B}`
..........................(13.2)
`color{blue} ✍️`If `P, V` and `T` are same, then `N` is also same for all gases. This is Avogadro’s hypothesis, that the number of molecules per unit volume is same for all gases at a fixed temperature and pressure.
`color{blue} ✍️`The number in 22.4 litres of any gas is `6.02 × 10^23`. This is known as Avogadro number and is denoted by NA. The mass of 22.4 litres of any gas is equal to its molecular weight in grams at S.T.P (standard temperature 273 K and pressure 1 atm).
`color{blue} ✍️`This amount of substance is called a mole (see Chapter 2 for a more precise definition). Avogadro had guessed the equality of numbers in equal volumes of gas at a fixed temperature and pressure from chemical reactions. Kinetic theory justifies this hypothesis.
`color{blue} ✍️`The perfect gas equation can be written as
`color{blue} {PV = μ RT}`
........................ (13.3)
`color{blue} ✍️`where `μ` is the number of moles and `R = N_A k_B` is a universal constant. The temperature `T` is absolute temperature. Choosing kelvin scale for absolute temperature, `R = 8.314 J mol^-1K^-1`.
Here
`color{blue} {mu = M/M_o = N/N_A}`
..............................(13.4)
`color{blue} ✍️`where `M` is the mass of the gas containing N molecules, `M_0` is the molar mass and `N_A` the Avogadro’s number. Using Eqs. (13.4) and (13.3) can also be written as
`color{purple} { PV = k_BNT} ` or `color{purple} {P = k_B nT}`
`color{blue} ✍️`where n is the number density, i.e. number of molecules per unit volume. `k_B` is the Boltzmann constant introduced above. Its value in SI units is `1.38 × 10^-23 J K^-1`.
Another useful form of Eq. (13.3) is
`color{blue} {P =(rhoRT)/M_o}`
..........................(13.5)
`color{blue} ✍️`where `ρ` is the mass density of the gas.
`color{blue} ✍️`A gas that satisfies Eq. (13.3) exactly at all pressures and temperatures is defined to be an `color{green} "ideal gas"`. An ideal gas is a simple theoretical model of a gas. No real gas is truly ideal. Fig. 13.1 shows departures from ideal gas behaviour for a real gas at three different temperatures.
`color{blue} ✍️`Notice that all curves approach the ideal gas behaviour for low pressures and high temperatures.
`color{blue} ✍️`At low pressures or high temperatures the molecules are far apart and molecular interactions are negligible. Without interactions the gas behaves like an ideal one.
`color{blue} ✍️`If we fix `μ` and `T` in Eq. (13.3), we get
`color{blue} {PV = "constant"`
.....................(13.6)
`color{blue} ✍️`i.e., keeping temperature constant, pressure of a given mass of gas varies inversely with volume.
`color{brown} bbul"Boyle’s law"`
`color{blue} ✍️`This is the famous `color{green} "Boyle’s law"`. Fig. 13.2 shows comparison between experimental P-V curves and the theoretical curves predicted by Boyle’s law. Once again you see that the agreement is good at high temperatures and low pressures.
`color{brown} bbul"Charles’ law"`
`color{blue} ✍️`Next, if you fix P, Eq. (13.1) shows that `V ∝ T` i.e., for a fixed pressure, the volume of a gas is proportional to its absolute temperature T (`color{green} "Charles’ law"`). See Fig. 13.3.
`color{blue} ✍️`Finally, consider a mixture of non-interacting ideal gases: `μ_1` moles of gas 1, `μ_2` moles of gas 2, etc. in a vessel of volume V at temperature T and pressure P. It is then found that the equation of state of the mixture is :
`color{blue} {PV = ( μ_1 + μ_2 +… ) RT}`
.................... (13.7)
i.e.
`color{blue} {P = mu_1 (RT)/V + mu_2 (RT)/V + ....}`
............(13.8)
`color{blue} { = P_1 + P_2 + .....}`
..................(13.9)
`color{blue} ✍️`Clearly `color{purple} {P_1 = μ_1 R T//V}` is the pressure gas 1 would exert at the same conditions of volume and temperature if no other gases were present. This is called the partial pressure of the gas.
`color{blue} ✍️`Thus, the total pressure of a mixture of ideal gases is the sum of partial pressures. This is Dalton’s law of partial pressures.
`color{blue} ✍️`We next consider some examples which give us information about the volume occupied by the molecules and the volume of a single molecule.
`color{blue} ✍️`Properties of gases are easier to understand than those of solids and liquids. This is mainly because in a gas, molecules are far from each other and their mutual interactions are negligible except when two molecules collide.
`color{blue} ✍️`Gases at low pressures and high temperatures much above that at which they liquefy (or solidify) approximately satisfy a simple relation between their pressure, temperature and volume given by (see Ch. 11) for a given sample of the gas.
`color{blue} {PV = KT}`
........... (13.1)
`color{blue} ✍️`Here `T` is the temperature in kelvin or (absolute) scale. K is a constant for the given sample but varies with the volume of the gas.
`color{blue} ✍️`If we now bring in the idea of atoms or molecules then K is proportional to the number of molecules, (say) N in the sample. We can write `K = N k` . Observation tells us that this k is same for all gases. It is called Boltzmann constant and is denoted by `k_B`.
As
`color {blue} {(P_1V_1)/(N_1T_1) = (P_2V_2)/(N_2T_2) =} " constant "color{blue} { = k_B}`
..........................(13.2)
`color{blue} ✍️`If `P, V` and `T` are same, then `N` is also same for all gases. This is Avogadro’s hypothesis, that the number of molecules per unit volume is same for all gases at a fixed temperature and pressure.
`color{blue} ✍️`The number in 22.4 litres of any gas is `6.02 × 10^23`. This is known as Avogadro number and is denoted by NA. The mass of 22.4 litres of any gas is equal to its molecular weight in grams at S.T.P (standard temperature 273 K and pressure 1 atm).
`color{blue} ✍️`This amount of substance is called a mole (see Chapter 2 for a more precise definition). Avogadro had guessed the equality of numbers in equal volumes of gas at a fixed temperature and pressure from chemical reactions. Kinetic theory justifies this hypothesis.
`color{blue} ✍️`The perfect gas equation can be written as
`color{blue} {PV = μ RT}`
........................ (13.3)
`color{blue} ✍️`where `μ` is the number of moles and `R = N_A k_B` is a universal constant. The temperature `T` is absolute temperature. Choosing kelvin scale for absolute temperature, `R = 8.314 J mol^-1K^-1`.
Here
`color{blue} {mu = M/M_o = N/N_A}`
..............................(13.4)
`color{blue} ✍️`where `M` is the mass of the gas containing N molecules, `M_0` is the molar mass and `N_A` the Avogadro’s number. Using Eqs. (13.4) and (13.3) can also be written as
`color{purple} { PV = k_BNT} ` or `color{purple} {P = k_B nT}`
`color{blue} ✍️`where n is the number density, i.e. number of molecules per unit volume. `k_B` is the Boltzmann constant introduced above. Its value in SI units is `1.38 × 10^-23 J K^-1`.
Another useful form of Eq. (13.3) is
`color{blue} {P =(rhoRT)/M_o}`
..........................(13.5)
`color{blue} ✍️`where `ρ` is the mass density of the gas.
`color{blue} ✍️`A gas that satisfies Eq. (13.3) exactly at all pressures and temperatures is defined to be an `color{green} "ideal gas"`. An ideal gas is a simple theoretical model of a gas. No real gas is truly ideal. Fig. 13.1 shows departures from ideal gas behaviour for a real gas at three different temperatures.
`color{blue} ✍️`Notice that all curves approach the ideal gas behaviour for low pressures and high temperatures.
`color{blue} ✍️`At low pressures or high temperatures the molecules are far apart and molecular interactions are negligible. Without interactions the gas behaves like an ideal one.
`color{blue} ✍️`If we fix `μ` and `T` in Eq. (13.3), we get
`color{blue} {PV = "constant"`
.....................(13.6)
`color{blue} ✍️`i.e., keeping temperature constant, pressure of a given mass of gas varies inversely with volume.
`color{brown} bbul"Boyle’s law"`
`color{blue} ✍️`This is the famous `color{green} "Boyle’s law"`. Fig. 13.2 shows comparison between experimental P-V curves and the theoretical curves predicted by Boyle’s law. Once again you see that the agreement is good at high temperatures and low pressures.
`color{brown} bbul"Charles’ law"`
`color{blue} ✍️`Next, if you fix P, Eq. (13.1) shows that `V ∝ T` i.e., for a fixed pressure, the volume of a gas is proportional to its absolute temperature T (`color{green} "Charles’ law"`). See Fig. 13.3.
`color{blue} ✍️`Finally, consider a mixture of non-interacting ideal gases: `μ_1` moles of gas 1, `μ_2` moles of gas 2, etc. in a vessel of volume V at temperature T and pressure P. It is then found that the equation of state of the mixture is :
`color{blue} {PV = ( μ_1 + μ_2 +… ) RT}`
.................... (13.7)
i.e.
`color{blue} {P = mu_1 (RT)/V + mu_2 (RT)/V + ....}`
............(13.8)
`color{blue} { = P_1 + P_2 + .....}`
..................(13.9)
`color{blue} ✍️`Clearly `color{purple} {P_1 = μ_1 R T//V}` is the pressure gas 1 would exert at the same conditions of volume and temperature if no other gases were present. This is called the partial pressure of the gas.
`color{blue} ✍️`Thus, the total pressure of a mixture of ideal gases is the sum of partial pressures. This is Dalton’s law of partial pressures.
`color{blue} ✍️`We next consider some examples which give us information about the volume occupied by the molecules and the volume of a single molecule.