A 10m height cylinder with cross sectional area of 0.1 m2 has a massless piston at the bottom as shown in Figure Q2. Water is at the top of the piston with air at 20oC over it. Under the piston there is air at 300K. The volume of the air is 0.3m3. Heat is now transferred to the air so that the piston moves up, spilling the water out over the sides of the cylinder. The specific volume of water is 0.001002m3/kg. Assume that the specific heat of air is constant and equal to 0.7 kJ/kg.K.
(Q) Find the pressure Po of the air over the water, the mass of water at the start of the process.
Why did the water get a job promotion? Because it was well-pressed! Now, let's get to your question.
We need to find the pressure Po of the air over the water and the mass of water at the start of the process. To do this, we can use the ideal gas law and the equation for specific volume.
First, let's find the pressure Po of the air over the water. We know that the volume of the air is 0.3m3 and the specific volume of water is 0.001002m3/kg. This means that for every kilogram of water, we have an additional volume of 0.001002m3. So, the total volume of air and water combined is 0.3m3 + 0.001002m3 = 0.301002m3.
Now, using the ideal gas law, we have:
PV = nRT
where P is the pressure, V is the volume, n is the number of moles, R is the gas constant, and T is the temperature.
Since the piston moves up and spills the water out over the sides of the cylinder, the volume remains constant. This means that the volume of the air and water combined is still 0.301002m3.
Rearranging the ideal gas law to solve for pressure, we have:
P = nRT / V
We can rearrange the equation further to solve for n, the number of moles:
n = PV / RT
Substituting the values we have, P = Po + Pg (where Pg is the pressure due to the weight of the water column):
n = (Po + Pg) V / RT
Now, let's find the mass of water at the start of the process. We know that the specific volume of water is 0.001002m3/kg. This means that for every kilogram of water, we have a volume of 0.001002m3. So, the mass of water (m) can be calculated as:
m = V / specific volume of water
Substituting the values we have, m = 0.301002m3 / 0.001002m3/kg = 300kg.
To summarize, the pressure Po of the air over the water is given by P = nRT / V, and the mass of water at the start of the process is 300kg. Now, that's a lot of water!
To find the pressure Po of the air over the water, we can use the ideal gas law:
PV = nRT
Where:
P = Pressure (in Pa)
V = Volume (in m^3)
n = Number of moles of gas
R = Ideal gas constant (8.314 J/mol.K)
T = Temperature (in Kelvin)
First, let's calculate the number of moles of air under the piston. We know the volume (V) and the temperature (T), so we can rearrange the equation:
n = PV / RT
Given:
V = 0.3 m^3
T = 300 K
R = 8.314 J/mol.K
Let's calculate n:
n = (0.3 m^3 × Po Pa) / (8.314 J/mol.K × 300 K)
Since we want the mass of water at the start of the process, we need to determine the amount of water present in the cylinder.
The volume of the air under the piston is equal to the volume of the cylinder minus the volume of the water. So, we can calculate the volume of the water as follows:
Volume of water = Volume of cylinder - Volume of air
Given:
Cross-sectional area of the cylinder = 0.1 m^2
Height of the cylinder = 10 m
Volume of cylinder = Cross-sectional area × Height
Volume of water = (Cross-sectional area × Height) - Volume of air
Volume of water = (0.1 m^2 × 10 m) - 0.3 m^3
Now, we can calculate the mass of water using the specific volume of water:
Mass of water = Volume of water / Specific volume of water
Given:
Specific volume of water = 0.001002 m^3/kg
Mass of water = (0.1 m^2 × 10 m - 0.3 m^3) / 0.001002 m^3/kg
The pressure Po of the air over the water and the mass of water at the start of the process can be found by substituting the calculated values into the respective formulas.
To find the pressure Po of the air over the water, we can use the ideal gas law equation, which states that PV = nRT.
First, we need to calculate the number of moles of air (n) under the piston. We can use the ideal gas law equation, PV = nRT, where P is the pressure, V is the volume, n is the number of moles, R is the universal gas constant, and T is the temperature.
Given:
- Volume of air (V) = 0.3 m^3
- Temperature of air (T) = 300 K
The universal gas constant (R) is approximately 8.314 J/(mol·K).
Rearranging the ideal gas law equation, we can calculate the number of moles:
n = PV / RT
n = (P * 0.3) / (8.314 * 300)
Now, we need to find the mass of water at the start of the process. The mass of an object is calculated by dividing its weight by the acceleration due to gravity (g).
Given:
- Height of the cylinder (h) = 10 m
- Cross-sectional area of the cylinder (A) = 0.1 m^2
- Specific volume of water (v) = 0.001002 m^3/kg
- Density of water (ρ) = 1/v
ρ = 1 / 0.001002
Now, calculate the mass of water:
mass of water = ρ * A * h
Finally, we can find the pressure Po of the air over the water using Pascal's law, which states that pressure is force divided by area.
The force (F) is equal to the weight of the water:
F = mass of water * g
Pressure (Po) = F / A
By following these steps, you can find the pressure Po of the air over the water and the mass of water at the start of the process.