gaseous state

basic assumptions of the kinetic theory

1. the gas consists of particles of negligible volume compared to the volume of the container it occupies

2. the gas particles exert no attractive forces on each other

3. the collisions between gas particles are perfectly elastic and no kinetic energy is lost on collision

4. the gas particles are in continuous random motion

5. the average kinetic energy of the gas particles is directly proportional to the absolute temperature (in Kelvin, K)

measuring standards of gas samples

• physical properties of gases vary depending on several factors ⇒ physical conditions have to be specified when measurements are made

• molar volume is the volume occupied by 1 mol of an ideal gas

  • s.t.p. at 10⁵ Pa [1 bar] and 273 K [0 ℃] → 22.7 dm³mol^-1

  • r.t.p. at 101325 Pa [1 atm] and 293 K [20 ℃] → 24 dm³mol^-1

• gases are usually described through the use of the four properties

  • pressure

  • volume

  • temperature

  • number of moles of gas, n

• the gas laws describe the relationships between pairs of the properties when the other two properties are kept constant

gas laws

• describe the behaviour of any gaseous substance, regardless of its identity

• combining the three gas laws ⇒ V is directly proportional to nT/p [V ∝ nT/p]

boyle’s law

states that at constant temperature, the volume of a fixed mass of gas is inversely proportional to its pressure

• V ∝ 1/p ⇒ pV = constant

• an increase in pressure will decrease its volume

• fixed mass ⇒ gas has a fixed amount ⇒ n is constant

charles’ law

states that at constant pressure, the volume of a fixed mass of gas is directly proportional to its absolute temperature (in Kelvin, K)

• V ∝ T ⇒ V/T = constant

• an increase in the temperature of the gas will increase its volume

avogadro’s law

states that at constant temperature and pressure, equal volumes of gases contain equal number of gas particles

• V ∝ n

• an increase in the amount of gas will increase its volume

ideal gas equation

ideal gas equation: pV = nRT

• p: pressure of gas in Pa or N m^-2

• v: volume of gas in m³

• T: temperature in K

• n: amount of gas in mol

• R: molar gas constant 8.31 J K^-1 mol^-1 [found in data booklet]

conversion of units [gas equation]

• make sure all quantities are in their correct units before substitution

  • pressure: 1 atm = 101325 Pa, 1 bar = 10⁵ Pa

  • volume: 1 m³ = 10³ dm³ = 10⁶ cm³

  • temperature: absolute temp (K) = temp (℃) + 273

to determine relative molecular mass using ideal gas equation → application of ideal gas equation

• since n = amount of gas = mass/molar mass, and the numerical values of molar mass and Mr are the same ⇒ pV = mass x RT/Mr

Mr = (mass x RT)/pV [mass is in g]

• since density of gas, ρ = mass/V

Mr = (ρRT)/p [density is in g m^-3)

• to convert g cm^-3 to g m^-3, multiply by 10⁶

calculating gas variables under different physical conditions

• to perform calculations involving gases under two different sets of physical conditions, the strategy is to rearrange the ideal gas equation into a form that can be equated with another rearranged ideal gas equation

  1. make R the subject of the ideal gas equation: R = pV/nT

  2. since R is a constant, for a gas that is first subjected to an initial set of physical conditions before a final set of physical conditions, the two sets of pV/nT can be equated: p_1V_1/n_1T_1 = p_2V_2/n_2T_2

  3. when any of the four variables (p, V, n and T) is kept constant, the combined equation above can be further simplified

plot pV vs p at constant temperature (T) for a fixed amount (n) of gas

• y-axis: pV, x-axis: p

• constants: T, n and R

• pV = nRT (no further manipulation required)

• from above, pV = constant, which is in the form of y = c

• the plot is a horizontal line that cuts the y-axis at y = nRT

• as the pressure, p, of a gas increases, its volume decreases, such that the product, pV remains constant since the pressure, p, is inversely proportional to the volume, V

the plot of pV vs p (at constant T) is the same as the plot of pV vs V (at constant T)

graphical plots for a fixed amount of ideal gas → n is constant

1. sketch V vs p (at constant T)

   • equation: pV = nRT ⇒ V = nRT(1/p)

2. sketch V vs 1/p (at constant T)

   • equation: pV = nRT ⇒ V = nRT(1/p)

3. sketch V vs T (K) (at constant p)

   • equation: pV = nRT ⇒ V = (nR/p) T

4. sketch V vs T (°C) (at constant p)

   • equation: pV = nRT ⇒ V = (nR/p)(T(°C)+273) ⇒ V = nR/p(T(°C)) + nR/p(273)

5. plot pV vs V (at constant T)

   • equation: pV = nRT

   • since pV = constant, p and V will vary such that pV remains constant

general shapes of graphs

• use the mathematical equation for straight line: y = mx + c

  • y and x are values on the axes, m is the gradient and c is the y-intercept

1. horizontal straight line cutting the y axis at k

2. straight line with positive gradient cutting y axis at c

3. straight line with positive gradient passing through origin ⇒ directly proportional

4. hyperbola

deviation from ideal gas behaviour

• for an ideal gas under all conditions, pV = nRT ⇒ pV/nRT = 1

  • a plot of pV/nRT against p should be a horizontal line → constant value of 1

• however, an ideal gas (which obeys the ideal gas equation) does not exist in reality

  • the ideal gas model was constructed so that the physical quantities that describe a real gas can be approximated to a good enough degree of accuracy without having to involve mathematics in a rigorous manner

• while pV = nRT strictly applies to only ideal gases, when applied to a real gas, the values calculated are generally good enough estimates

real gases

• real gases do not obey the ideal gas equation

• deviations (positive or negative) from ideal gas behaviour can be attributed to these properties of real gases

  • significant attractive forces between the gas particles

  • significant volume of gas particles

• the extent and nature of deviation depends on

  • temperature

  • pressure

  • nature of the gas → strength of intermolecular forces and molecular size

negative deviation

real gases tend to exhibit negative deviation when the temperature is low [temperature: experimental temperature in Kelvin]

• the kinetic theory assumes that an ideal gas is made up of particles (atoms or molecules) with negligible intermolecular forces of attraction

• however, at low temperatures, gas particles move slowly and have low kinetic energy

• they have more time for interaction with each other and hence intermolecular forces of attraction between the particles are significant

• as particles collide with the walls of the container with lower frequency and with lesser force than at high temperatures

• the pressure exerted by a real gas is lower than that by an
  ideal gas

  • since p(real) < p(ideal) ⇒ p(real)V/RT < p(ideal)V/RT ⇒ p(real)V/RT < 1

negative deviation becomes greater when temperature lowers

always specify the type of intermolecular forces of attraction between the gas

when is the temperature considered low

the experimental temperature is considered low when it is near the boiling point of the gas → depends on the identity of the gas

summary for negative deviation

• negative deviation occurs at low temperature → relative to boiling point of gas

• extent of negative deviation depends on

  • experimental temperature ⇒ the lower the temperature, the greater the negative deviation

  • strength of intermolecular forces of attraction ⇒ the stronger the intermolecular forces of attraction, the greater the extent of negative deviation

positive deviation

real gases tend to exhibit positive deviation when the pressure is high

• the kinetic theory assumes that an ideal gas is made up of particles (atoms or molecules) with negligible volume compared to the volume of the container

• at high pressures, the gas is compressed, and the total volume occupied by the gas is reduced

• gas particles become closer together, and the molecular size (volume) of the gas particles becomes significant when compared to the total volume of the gas (i.e. the volume of the container the gas is in)

• since V(real) > V(ideal) ⇒ pV(real)/RT > pV(ideal)/RT ⇒ pV(real)/RT > 1

  • the volume of a gas particle is determined by its electron cloud size and estimated by its Mr ⇒ a gas particle with a bigger Mr has a larger electron cloud size

summary of positive deviation

• positive deviation occurs at high pressure

• extent of positive deviation depends on the size of gas particles → indicated by the Ar or Mr

  • the greater the molecular size, the greater the extent of the positive deviation

    • if a gas has larger sized particles, at high pressures, the volume occupied by the gas particles is even more significant compared to the volume of the container the gas is in (in comparison to a gas with smaller sized particles)

    • this results in the volume of the real gas being much greater than the volume of an ideal gas ⇒ V(real) >>> V(ideal)

    • this would result in the pV/nRT being much larger than 1 [more positive deviation]

conditions necessary for gas to approach ideal gas behaviour

• real gases only tend towards ideal gas behaviour when

  • pressure is very low

  • temperature is very high

low pressure

• at very low pressures, gas molecules are further apart (intermolecular forces of attraction are also negligible)

• the volume of the gas molecules is insignificant compared to the volume of the container

high temperature

at high temperatures, gas molecules have sufficient kinetic energy to overcome intermolecular forces of attraction which can be considered insignificant

dalton’s law of partial pressure → definition

the total pressure exerted by a mixture of gases which do not react is equal to the sum of the partial pressures of the constituent gases at the same temperature

• only applies when the volume of the vessel of individual gases is the same as the volume of the gas mixture

dalton’s law of partial pressure

• mathematically expressed as: total pressure of the gas mixture, P_total = P₁ + P₂ + P₃ + … + P_i

  • P_i = partial pressure of gas i in the mixture = the pressure which gas i exerts if it alone occupies the container at the same temperature

• the total pressure of air is the sum of partial pressures of O₂(g), N₂(g), CO₂(g) and other gases present

  • P_total = P_O2 + P_N2 + P_CO2 + …

  • P_O2 is the pressure exerted by O₂ in the air mixture; it is also the pressure exerted by the same amount of O₂ if it were to exist alone without the other gases

relating partial pressure to total pressure:

• P_A/P_T = (n_ART/VT) / (n_TRT/VT) ⇒ P_A = (n_A/n_T) x P_T [n_A/n_T is the mole fraction of gas A in the gaseous mixture]