a 5.0 dm3 sample of oxygen at a pressure of 200 kPa and 2.0 dm3 sample of nitrogen at a pressure of 500 kPa are introduced into a 2.5 dm3 vessel.
what is the total pressure in the vessel?
There may be an easier way to work this problem but I would do the following:
Use PV = nRT and solve for n for oxygen.
Use PV = nRT and solve for n for nitrogen.
noxygen + nnitrogen = total n.
Then PV = nRT, substitute the new volume and solve for total P. There is no T listed in the problem but you can use any T you wish as long as the same T is used throughout.
To find the total pressure in the vessel, we need to consider the ideal gas law, which states that the product of pressure and volume divided by the product of moles and temperature is constant for an ideal gas.
1. First, let's calculate the number of moles for each gas sample using the ideal gas law equation:
For oxygen:
P₁V₁ = n₁RT
n₁ = (P₁V₁) / (RT)
where:
P₁ = 200 kPa (pressure of oxygen)
V₁ = 5.0 dm³ (volume of oxygen)
R = 8.314 J/(mol·K) (universal gas constant)
Substitute the values into the equation:
n₁ = (200 kPa * 5.0 dm³) / (8.314 J/(mol·K) * T)
For nitrogen:
P₂V₂ = n₂RT
n₂ = (P₂V₂) / (RT)
where:
P₂ = 500 kPa (pressure of nitrogen)
V₂ = 2.0 dm³ (volume of nitrogen)
Substitute the values into the equation:
n₂ = (500 kPa * 2.0 dm³) / (8.314 J/(mol·K) * T)
2. Since the two gas samples are introduced into a common vessel, we can assume that the moles in each sample remain constant, meaning n₁ = n₂ = n.
3. Now, we can calculate the total number of moles in the vessel, n_total:
n_total = n₁ + n₂
4. Finally, using the total number of moles, we can find the total pressure, P_total, in the vessel by rearranging the ideal gas law equation:
P_total = (n_total * R * T) / V_total
where:
V_total = 2.5 dm³ (volume of the vessel)
Substitute the values into the equation to calculate P_total.
To find the total pressure in the vessel, we need to use the ideal gas law, which states that the pressure (P) of a gas is directly proportional to its volume (V) and the number of moles (n) of gas present, and inversely proportional to the temperature (T) of the gas. The equation is given as:
PV = nRT
Where:
P = pressure
V = volume
n = number of moles
R = ideal gas constant
T = temperature
In this case, we are given the volumes of the oxygen and nitrogen samples and their respective pressures. However, we do not know the temperature or the number of moles of each gas.
To calculate the total pressure, we need to first determine the number of moles of each gas. To do this, we can use the ideal gas equation rearranged to solve for n:
n = PV / RT
Using the given information for each gas, we can calculate the number of moles:
For oxygen:
P_oxygen = 200 kPa
V_oxygen = 5.0 dm3
For nitrogen:
P_nitrogen = 500 kPa
V_nitrogen = 2.0 dm3
We can use the ideal gas constant, R, which is 8.314 J/(mol·K). However, since the pressure is in kilopascals (kPa) and the volume is in cubic decimeters (dm3), we need to convert them to their respective SI units:
1 kPa = 1000 Pa
1 dm3 = 0.001 m3
Therefore, we have:
P_oxygen = 200,000 Pa
V_oxygen = 0.005 m3
P_nitrogen = 500,000 Pa
V_nitrogen = 0.002 m3
Now we can calculate the number of moles of each gas:
n_oxygen = (P_oxygen * V_oxygen) / (R * T)
n_nitrogen = (P_nitrogen * V_nitrogen) / (R * T)
Since the temperature is not given, it is safe to assume that the temperature remains constant throughout the process. Therefore, we can ignore the temperature term and compare the relative pressures to find the total pressure.
To do this, we divide each pressure by its respective volume, keeping the units consistent:
P_total = (P_oxygen + P_nitrogen) / V_vessel
V_vessel = 2.5 dm3
Now we can substitute the values into the equation to find the total pressure:
P_total = (200,000/0.005 + 500,000/0.002) / 2.5
Simplifying the equation:
P_total = (40,000,000 + 250,000,000) / 2.5
P_total = 290,000,000 / 2.5
P_total = 116,000,000 Pa
Therefore, the total pressure in the vessel is 116,000,000 Pa.