Three bulbs are connected by tubing, and the tubing is evacuated. The volume of the tubing is 27.0 mL. The first bulb has a volume of 37.0 mL and contains 5.65 atm of argon, the second bulb has a volume of 250 mL and contains 2.16 atm of neon, and the third bulb has a volume of 48.0 mL and contains 8.93 atm of hydrogen. If the stopcocks (valves) that isolate all three bulbs are opened, what is the final pressure of the whole system in atm?
A multistep problem but simple to do.
Use P1V1 = P2V2. For example, calculate the total volume by adding the volumes of each bulb. Don't forget to add in the volume of the tubing. Bulb 1 is 37.0 mL with P of 5.65 atm.
So P1 is 5.65, V1 is 37.0 mL, V2 is the total volume and you calculate P2 which the partial pressure of Ar under the new conditions. Do that for each bulb, then add the partial pressures (the P2 values) of each gas to find the total pressure of the system.
To find the final pressure of the whole system, we can use the principle of Dalton's Law of Partial Pressures. According to this principle, the total pressure exerted by a mixture of ideal gases is equal to the sum of the partial pressures of each individual gas.
The partial pressure of each gas is given by the product of the gas's mole fraction and the total pressure. The mole fraction of each gas can be calculated by dividing the moles of each gas by the total moles.
First, we need to find the moles of each gas in the system. To do this, we can use the ideal gas equation:
PV = nRT,
where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant, and T is the temperature (which is assumed to be constant in this case).
For argon:
n₁ = (P₁ * V₁) / (R * T)
For neon:
n₂ = (P₂ * V₂) / (R * T)
For hydrogen:
n₃ = (P₃ * V₃) / (R * T)
Given values:
P₁ = 5.65 atm
V₁ = 37.0 mL = 0.037 L
P₂ = 2.16 atm
V₂ = 250 mL = 0.25 L
P₃ = 8.93 atm
V₃ = 48.0 mL = 0.048 L
R = 0.0821 L·atm/(mol·K) (ideal gas constant)
T = constant
Calculating the moles:
n₁ = (5.65 * 0.037) / (0.0821 * T)
n₂ = (2.16 * 0.25) / (0.0821 * T)
n₃ = (8.93 * 0.048) / (0.0821 * T)
Next, we need to calculate the total moles of the system:
n_total = n₁ + n₂ + n₃
Now, to find the mole fractions:
X₁ = n₁ / n_total
X₂ = n₂ / n_total
X₃ = n₃ / n_total
The partial pressures of each gas are given by:
P₁_partial = X₁ * P_total
P₂_partial = X₂ * P_total
P₃_partial = X₃ * P_total
Since the total pressure is the sum of all the partial pressures:
P_total = P₁_partial + P₂_partial + P₃_partial
Substituting the values, we can calculate the final pressure.
Final pressure = P_total
Note: The given values for the pressure are already in atm, so no conversion is necessary.
Let's assume the temperature (T) is a constant value. If you have the value of T, please provide it to calculate the final pressure.
To find the final pressure of the whole system, we need to consider the ideal gas law, which states:
PV = nRT
Where:
- P is the pressure of the gas
- V is the volume of the gas
- n is the number of moles of the gas
- R is the ideal gas constant
- T is the temperature of the gas in kelvin
Since there is no change in temperature, we can ignore it for this problem. Therefore, we can rewrite the equation as:
PV = nR
To find the final pressure of the whole system, we need to consider the individual pressures and volumes of each bulb. Let's calculate the total volume and total moles first:
Total volume = Volume of the tubing + Volume of bulb 1 + Volume of bulb 2 + Volume of bulb 3
= 27.0 mL + 37.0 mL + 250 mL + 48.0 mL
= 362.0 mL
To convert the volume to liters (as the ideal gas constant has SI units), we divide by 1000:
Total volume = 362.0 mL ÷ 1000
= 0.362 L
Now let's consider the total moles of gas in each bulb:
Number of moles = Total pressure / Partial pressure of the gas
Number of moles of argon in bulb 1 = (5.65 atm * 37.0 mL) / (0.0821 L·atm/mol·K)
Number of moles of neon in bulb 2 = (2.16 atm * 250 mL) / (0.0821 L·atm/mol·K)
Number of moles of hydrogen in bulb 3 = (8.93 atm * 48.0 mL) / (0.0821 L·atm/mol·K)
Now let's calculate the total moles of gas:
Total moles = Number of moles of argon + Number of moles of neon + Number of moles of hydrogen
Finally, let's calculate the final pressure of the system:
Final pressure = Total moles * R / Total volume
Now you can plug in the values and calculate the final pressure of the whole system.