Question #651b6
1 Answer
Explanation:
!!! LONG ANSWER !!
The idea here is that you can use the ideal gas law equation to find the pressure in box
At that point, you can use Dalton's Law of Partial Pressures to find the total pressure in box
So, you know that you're dealing with two boxes of different volumes that contain different numbers of moles of gas.
So, the ideal gas law equation looks like this
#color(blue)(|bar(ul(color(white)(a/a)PV = nRTcolor(white)(a/a)|)))" "# , where
If you take
#P_A * V_A = n_A * RT" " " "color(orange)(("*")) -># for box#"A"#
Now, you know that
#V_B = 2 xx V_A -># the volume of box#"B"# is twice that of box#"A"#
and
#n_B = 2 xx n_A -># box#"B"# contains twice as many moles of gas as box#"A"#
You can thus use the ideal gas law equation to write
#P_B * V_B = n_B * RT -># for box#"B"#
But this is equivalent to
#P_B * (2V_A) = (2n_A) * RT#
As you can see, you can use equation
#P_B * color(red)(cancel(color(black)(2)))V_A = color(red)(cancel(color(black)(2)))n_A * RT#
#{(P_B * V_A = n_A * RT), (P_A * V_A = N_A * RT):}#
the pressure in box
#P_B = P_A = "434.0 torr"#
At this point, you can use Boyle's Law to find the pressure exerted by the gas in box
As you know, volume and pressure have an inverse relationship when temperature and number of moles are kept constant.
#color(blue)(|bar(ul(color(white)(a/a)P_1V_1 = P_2V_2color(white)(a/a)|)))" "# , where
In your case, you have
#P_A * color(red)(cancel(color(black)(V_A))) = P_("A in box B") * (2color(red)(cancel(color(black)(V_A))))#
This will get you
#P_("A in box B") = P_A/2 = "434.0 torr"/2 = "217.0 torr"#
So, the moles of methane will exert a pressure of
According to Dalton's Law of Partial Pressures, the total pressure of a gaseous mixture will be equal to the sum of the partial pressures of each constituent of the mixture if left alone in the same volume at the same temperature as the mixture.
You will thus have
#P_"total" = P_B + P_("A in box B")#
This will get you
#P_"total" = "434.0 torr" + "217.0 torr" = color(green)(|bar(ul(color(white)(a/a)"651.0 torr"color(white)(a/a)|)))#
ALTERNATIVE APPROACH
Here's an alternative method to use to find the total pressure of the mixture.
According to Dalton's Law of Partial Pressures, the partial pressure of a gas that's part of a gaseous mixture depends on its mole fraction and on the total pressure of the mixture.
#color(blue)(|bar(ul(color(white)(a/a)P_"gas i" = chi_"gas i" xx P_"total"color(white)(a/a)|)))#
Here
Now, we've established that the gas in box
#chi_B = "number of moles of helium"/"total number of moles of gas"#
#chi_B = (8 color(red)(cancel(color(black)("moles"))))/((8 + 4)color(red)(cancel(color(black)("moles")))) = 2/3#
This means that you have
#P_B = chi_B xx P_"total"#
and therefore
#P_"total" = P_B /chi_B#
This once again gets you
#P_"total" = "434.0 torr"/(2/3) = color(green)(|bar(ul(color(white)(a/a)"651.0 torr"color(white)(a/a)|)))#