A flask contains three times as many moles as #H_2# as it does #O_2# gas. If hydrogen and oxygen are the only present gases, what is the total pressure flask if the partial pressure due to oxygen is #P#?

According to Dalton's Law of Partial Pressures, the total pressure of a gaseous mixture will be equal to the individual partial pressure of each component of the mixture.

Simply put, to get the total pressure of a gaseous mixture, you add up the partial pressure you'd get for each component of that mixture if it occupied the volume of the mixture alone.

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Mathematically, this can written as

#color(blue)(|bar(ul(color(white)(a/a)P_"total" = sum_i P_icolor(white)(a/a)|)))" "#, where

#P_"total"# - the total pressure of the gaseous mixture
#P_i# - the partial pressure of component #i# of said mixture

In your case, the mixture is said to contain hydrogen gas, #"H"_2#, and oxygen gas, #"O"_2#, in a #3:1# mole ratio.

This means that if you take #n_(O_2)# to be the number of moles of oxygen gas present in the mixture, you can say that

#n_(H_2) = 3 xx n_(O_2)" "color(purple)("(*)")#

Now, let's assume that the mixture is being held in a container that has a volume of #V# liters. If you take #P# to be the partial pressure of oxygen gas, you can use the ideal gas law equation

#color(blue)(|bar(ul(color(white)(a/a)PV = nRT color(white)(a/a)|)))#

to express #V# in terms of #P# and #n_(O_2)#. You will thus have

#P * V = n_(O_2) * RT implies V = n_(O_2)/P * RT" "color(red)("(*)")#

Now, you can write the same thing for #P^'#, the partial pressure of hydrogen gas in the same volume #V#

#P^' * V = n_(H_2) * RT implies V = n_(H_2)/P^' * RT#

Use equation #color(red)("(*)")# to write

#n_(O_2)/P * color(red)(cancel(color(black)(RT))) = n_(H_2)/P^' * color(red)(cancel(color(black)(RT)))#

Rearrange to find #P^'#, the partial pressure of hydrogen gas

#P^' = n_(H_2)/n_(O_2) * P#

Use equation #color(purple)("(*)")# to write

#P^' = (3 xx color(red)(cancel(color(black)(n_(O_2)))))/color(red)(cancel(color(black)(n_(O_2)))) * P#

#P^' = 3 xx P#

Finally, plug this into the equation that describes Dalton's Law of Partial Pressures to get the total pressure of the mixture, #P_"total"#

#P_"total" = P^' + P#

#P_"total" = 3 xx P + P = color(green)(|bar(ul(color(white)(a/a)4 xx Pcolor(white)(a/a)|)))#