Calculate charles's law/Gay's' law for Pressure/volume and temperature

Calculate charles's law/Gay's' law for Pressure/volume and temperature 

solution

The particle theory of gas pressure was explained in Part 1 so this section concentrates on the gas law calculations involving pressure and volume and their variation with temperature.

• Charles's/Gay–Lussac's Law states that for a fixed mass of gas ...
• (i) the volume of a gas is directly proportional to the absolute temperature (K) at constant pressure

§  V = constant x T (right graph), or

§  V/T = constant, or

§  V₁/V₂ = T₁/T₂  for conditions changing from 1 (initial) to 2 (final),

§  or V₁/T₁ = V₂/T₂  for constant pressure

§  V₁ x T₂ = V₂ x T₁

§  V₂ = V₁ x T₂/T₁

§  or T₂ = T₁ x V₂/V₁

§  Kinetic particle reasoning - increasing the temperature increases the kinetic energy of the molecules giving more forceful collisions which push out (expand) the gas at constant pressure.

§  Note that the graphs extrapolate back to 0K (absolute zero, Kelvin scale) or -273^oC (Celsius scale).

• OR (ii) the pressure of a gas is directly proportional to the absolute temperature (K) at constant volume,
  
  
  

§  p = constant x T (right graph), or

§  p/T = constant, or

§  p₁/p₂ = T₁/T2  for conditions changing from 1 (initial) to 2 (final),

§  or p₁/T₁ = p₂/T₂  for constant volume

§  p₁ x T₂ = T₁ x p₂

§  p₂ = p₁ x T₂/T₁

§  or T₂ = T₁ x p₂/p₁

§  Kinetic particle reasoning - increasing the temperature increases the kinetic energy of the molecules giving more forceful collisions that increase the pressure if the volume is constrained (kept constant).

§  Note again that the graphs extrapolate back to 0K (absolute zero, Kelvin scale) or -273^oC (Celsius scale).

• In all calculations, the absolute or Kelvin scale of temperature must be used for T (K = ^oC + 273).
• If all the laws described in 4a and 4b are combined, you get the following general expression
• p x V/T = a constant (for a given mass of gas).
• This can be expressed in generalised form for calculations based on an initial set of conditions₁ (1) changing to a new and final set of conditions₂ (2) for a given mass of gas, giving the combined pressure–volume–temperature gas calculation equation ...
•  

p1 x V1		p2 x V2
––––––––––––	=	––––––––––––
T1		T2

• In shorthand': p₁V₁ /T ₁ = p₂V₂ / T₂
• therefore the three permutations for problem solving involving all three variables are:
• p₂ = p₁V₁T₂ / V₂T₁
• V₂ = p₁V₁T₂ / p₂T₁
• T₂ = p₂V₂T₁ / p₁V₁
• If one variable is constant, the permutations for the other two variables are:
• p₂ = p₁V₁ / V₂   (at constant temperature)
• V₂ = V₁T₂ / T₁   (at constant pressure)
• T₂ = p₂T₁ / p₁   (at constant volume)
• Note:
• If the temperature is constant you get Boyle's Law.
• If p or V is constant you get Charles's/Gay–Lussac's Law.
• You can use any volume or pressure units you like as long as both p's or both V's have the same units.
• The graphs of p or V versus temperature become invalid once the gas has condensed into a liquid BUT when extrapolated back all the lines seem to originate from y = 0 (for p or V), x = –273^oC (for T).
• This was part of the scientific evidence that led to the belief that –273^oC was the lowest possible temperature, though there is no theoretical upper limit at all.
• This led to the devising of a new thermodynamic absolute temperature scale or Kelvin scale which starts at OK.

§  e.g. ice melts at 0^oC or 273K and water boils at 100^oC or 373K.