2 spherical containers- P and Q are connected by a tap. P contains a gas at pressure 5 atm at 300K. Q contains same gas at pressure 2 atm at 400K. If tap is opened find the final pressure. Given: volume of Q is 4 times volume of P.
To find the final pressure when the tap is opened, we can use the ideal gas law, which states:
PV = nRT
Where:
P = pressure
V = volume
n = number of moles of gas
R = ideal gas constant
T = temperature
First, let's calculate the number of moles of gas in containers P and Q separately.
For container P:
Given pressure, P1 = 5 atm
Given temperature, T1 = 300 K
Given volume, V1
For container Q:
Given pressure, P2 = 2 atm
Given temperature, T2 = 400 K
Given volume, V2 = 4V1 (since the volume of Q is 4 times the volume of P)
Now, let's calculate the number of moles of gas in each container using the ideal gas law:
n1 = (P1 * V1) / (R * T1) -- (1)
n2 = (P2 * V2) / (R * T2) -- (2)
Next, when the tap is opened, the gas in containers P and Q will mix. The total number of moles of gas after mixing can be calculated as:
ntotal = n1 + n2
Now, let's find the final pressure, Pfinal. We can rearrange the ideal gas law equation as follows:
Pfinal = (ntotal * R * Tfinal) / Vfinal -- (3)
Since the volumes of P and Q are connected by the tap, we know that Vfinal = V1 + V2. Substituting this value and rearranging equation (3), we can solve for Pfinal:
Pfinal = (ntotal * R * Tfinal) / (V1 + V2) -- (4)
Substituting equations (1) and (2) into equation (4), we get:
Pfinal = ((P1 * V1) / (R * T1) + (P2 * V2) / (R * T2)) * (R * Tfinal) / (V1 + V2)
Now, substitute the given values and solve for Pfinal:
Pfinal = ((5 atm * V1) / (R * 300 K) + (2 atm * 4V1) / (R * 400 K)) * (R * Tfinal) / (V1 + 4V1)
Simplifying further:
Pfinal = ((5 / (R * 300)) + (2 * 4 / (R * 400))) * (R * Tfinal) / (1 + 4)
Pfinal = ((5 / (R * 300)) + (8 / (R * 400))) * (R * Tfinal) / 5
Pfinal = (5 / (R * 300) + 8 / (R * 400)) * Tfinal
Finally, since the equation already contains the ideal gas constant (R), we can simplify further to get:
Pfinal = (5/300 + 8/400) * Tfinal
Pfinal = (1/60 + 1/50) * Tfinal
Pfinal = (50 + 60) / (60 * 50) * Tfinal
Pfinal = (110 / 3000) * Tfinal
Therefore, the final pressure, Pfinal, is equal to (110 / 3000) * Tfinal, where Tfinal is the final temperature when the tap is opened.