Chapter 5, Problem 5.68P

You have been assigned the task of measuring the equilibrium constant for the reaction ${\text{N}}_2{\text{O}}_4\rightleftharpoons {\text{2NO}}_2$; as a function of temperature. To do so, you evacuate a rigid 2-liter vessel equipped with a pressure gauge, ?ll the vessel with a mixture of NO₂and N₂O₄, and then heat it to T₀= 473 K, a temperature at which you know the gas is essentially pure NO₂. The gauge pressure at this point is noted to be 1.00 atm. You then decrease the temperature in stages, recording the equilibrium gauge pressure at each temperature. The data are as follows:

(a) How many gram-moles of NO₂are in the vessel at 473 K?

(b) The reaction equilibrium constant is

   $K_{\text{p}}=p^2{}_{{\text{NO}}_{\text{2}}}/p_{{\text{N}}_{\text{2}}{\text{O}}_{\text{4}}}$

where $p_{{\text{NO}}_{\text{2}}}$and $p_{{\text{N}}_{\text{2}}{\text{O}}_{\text{4}}}$are the equilibrium partial pressures of NO₂and N₂O₄. Derive an equation or a series of equations for calculating K_p(atm) from speci?ed values of T and P_gauge. (Suggestion: Begin by de?ning n₁and n₂as the moles of NO₂and N₂O₄present at equilibrium.) Then calculate K_pfor T = 350 K, 335 K, 315 K, and 300 K (Suggestion: Use a spreadsheet.)

(c) The equilibrium constant should vary with temperature according to the relation

   $K_p=a{e}^{-b/T}$

Use the results of Part (b) to determine the values of a and b by curve-?tting. [Suggestion: Use the spreadsheet from Pan (b).]