To tackle these two questions, we need to apply principles from thermodynamics and fluid mechanics. Let's break them down step by step.
1. Heat Transfer Required for the Piston to Rise
In this scenario, we have a piston-cylinder device containing helium gas. The initial conditions are as follows:
- Mass of helium, m = 0.5 kg
- Initial pressure, P1 = 100 kPa
- Initial temperature, T1 = 25 °C (which is 298 K)
- Pressure required to lift the piston, P2 = 500 kPa
To find the heat that must be transferred to the helium before the piston starts rising, we can use the ideal gas law and the concept of work done on/by the gas.
Step 1: Calculate the Initial Volume of Helium
Using the ideal gas law, PV = nRT, we can rearrange it to find the volume (V) of the gas:
V = nRT/P
First, we need to find the number of moles (n) of helium:
Helium has a molar mass of approximately 4 g/mol, so:
n = m / M = 0.5 kg / 0.004 kg/mol = 125 moles
Now, substituting the values into the ideal gas equation:
V = (125 mol) * (8.314 J/(mol·K)) * (298 K) / (100,000 Pa) = 3.1 m³
Step 2: Determine the Work Done to Raise the Piston
The work done on the gas when the piston rises can be calculated using the formula:
W = PΔV
However, since we are looking for the heat transfer (Q) required to raise the pressure from 100 kPa to 500 kPa, we can use the first law of thermodynamics:
ΔU = Q - W
Where ΔU is the change in internal energy. For an ideal gas, ΔU can be calculated as:
ΔU = nCvΔT
Where Cv for helium (a monatomic gas) is approximately 3/2 R. We need to find the final temperature (T2) when the pressure is 500 kPa.
Step 3: Calculate the Final Temperature
Using the ideal gas law again:
P2V = nRT2
Rearranging gives us:
T2 = P2V / (nR)
Substituting the known values:
T2 = (500,000 Pa) * (3.1 m³) / (125 mol * 8.314 J/(mol·K)) = 1875 K
Step 4: Calculate the Change in Internal Energy
Now we can find ΔU:
ΔU = n * (3/2) * R * (T2 - T1)
ΔU = 125 mol * (3/2) * 8.314 J/(mol·K) * (1875 K - 298 K)
ΔU = 125 * 1.5 * 8.314 * 1577 = 1,000,000 J
Step 5: Calculate Heat Transfer
Assuming no work is done (since the piston is just about to move), we can set W = 0:
Q = ΔU = 1,000,000 J
Therefore, the heat that must be transferred to the helium before the piston starts rising is approximately 1,000,000 Joules.
2. Mass of the Petcock for Boiling at 120 °C
In this case, we need to determine the mass of the petcock that will allow steam to escape from a pressure cooker at a temperature of 120 °C, with an atmospheric pressure of 101.3 kPa.
Understanding the Pressure Conditions
The boiling point of water increases with pressure. At 120 °C, the saturation pressure of steam is approximately 124 kPa. This means that for water to boil at this temperature, the pressure inside the cooker must be at least 124 kPa.
Step 1: Calculate the Required Pressure Difference
The pressure inside the cooker must exceed the atmospheric pressure plus the saturation pressure:
P_inside = P_atm + P_saturation = 101.3 kPa + 124 kPa = 225.3 kPa
Step 2: Determine the Force Exerted by the Steam
The force exerted by the steam can be calculated using the area of the petcock opening:
F = P * A
Where A = 5 mm² = 5 x 10^-6 m². The pressure difference (P) is the difference between the internal pressure and atmospheric pressure:
P = 225.3 kPa - 101.3 kPa = 124 kPa = 124,000 Pa
Now, substituting the values:
F = 124,000 Pa * 5 x 10^-6 m² = 0.62 N
Step 3: Calculate the Mass of the Petcock
The mass of the petcock can be calculated using the force of gravity:
F = m * g
Where g = 9.81 m/s² (acceleration due to gravity). Rearranging gives us:
m = F / g = 0.62 N / 9.81 m/s² ≈ 0.063 kg
Thus, the mass of the petcock should be approximately 0.063 kg, or 63 grams, to allow boiling at 120 °C with an atmospheric pressure of 101.3 kPa.
