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This problem is providing us with the reaction whereby methane is produced from solid carbon and gaseous hydrogen, as well as its Kp and asks for the total pressure at equilibrium for a specified addition of reactants. At the end, the answer turns out to be 11.7 atm.
In chemistry, when a chemical reaction takes place, it proceeds until a point called equilibrium, where the concentration of all the species start becoming unchanged. Thus, we can use equilibrium expressions whenever we needed pressures or concentrations at equilibrium.
In this case, for the reaction we were given, we can write the following equilibrium expression, where the solid carbon is excluded:
[tex]Kp=\frac{p_{CH_4}}{p_{H_2}^2} =0.263[/tex]
Which can also be written in terms of the reaction extent and the initial pressure of hydrogen:
[tex]0.263=\frac{x}{(p_{H_2}^0-2x)^2}[/tex]
Thus, we calculate the initial pressure of hydrogen with the temperature and volume of the flask:
[tex]p_{H_2}^0=\frac{(4.50g/2.02g/mol)*0.08206\frac{atm*L}{mol*K}*1000K}{10.0L}\\ \\p_{H_2}^0=18.3atm[/tex]
Hence, we solve for x in the equilibrium expression as follows:
[tex]0.263=\frac{x}{(18.3-2x)^2}\\\\x_1=6.64atm\\\\x_2=12.6atm[/tex]
Where the correct value is 6.64 atm because the 12.6 atm turns out into a negative pressure for hydrogen at equilibrium. Finally, we calculate the total pressure with:
[tex]P_T=p_{CH_4}+p_{H_2}=x+18.3-2x\\\\P_T=6.64atm+18.3atm-2*6.64atm\\\\P_T=11.7atm[/tex]
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