a flask contains helium gas at 11k. the flask is thermally isolated and the gas is allowed to escape slowly through a narrow tube until the pressure within the flask is 0.98 atm and the temperature is 4.2K. if the gas is ideal, find the initial pressure for the flask to be entirely filled with liquid at the end of the process.
To find the initial pressure for the flask to be entirely filled with liquid at the end of the process, we need to determine the phase of the helium gas at the final conditions and then use the phase diagram of helium to find the corresponding pressure.
1. Determine the phase of helium gas at the final conditions:
At a temperature of 4.2K and a pressure of 0.98 atm, helium is in the liquid phase. This is because helium liquefies at very low temperatures and high pressures.
2. Use the phase diagram of helium to find the corresponding pressure:
The phase diagram for helium shows the temperature and pressure conditions at which helium exists in different phases. At 4.2K, we can see from the phase diagram that helium is in the liquid phase for pressures greater than approximately 0.95 atm.
Therefore, the initial pressure in the flask must be greater than 0.95 atm for it to be entirely filled with liquid at the end of the process. However, without additional information, we cannot determine the specific initial pressure required.
To find the initial pressure for the flask to be entirely filled with liquid at the end of the process, we need to determine the conditions under which helium gas can turn into a liquid.
We can use the phase diagram of helium to determine the temperature and pressure at which it transitions from a gas to a liquid. Helium is unique because it undergoes the phase transition to a liquid at extremely low temperatures.
The phase diagram of helium shows that at temperatures below 5.2K, helium exists as a liquid at any pressure.
Given that the temperature at the end of the process is 4.2K, which is below the critical temperature of helium, we can conclude that the helium gas in the flask will completely condense into a liquid.
Now we need to determine the initial pressure that results in a final pressure of 0.98 atm at 4.2K.
For an ideal gas, we can use the Ideal Gas Law:
PV = nRT
Where:
P = Pressure
V = Volume
n = Moles of gas
R = Ideal Gas Constant
T = Temperature
Since the flask is thermally isolated, the number of moles of gas (n) remains constant throughout the process. Thus, we can rewrite the ideal gas law as:
P₁V₁ = P₂V₂
Where:
P₁ = Initial pressure
V₁ = Initial volume
P₂ = Final pressure
V₂ = Final volume
Since the gas is escaping slowly through a narrow tube, we can assume that the volume of the flask remains constant. Therefore, V₁ = V₂.
Substituting the given values, we get:
P₁ * V₁ = 0.98 atm * V₁
Since V₁ = V₂, we can simplify the equation to:
P₁ = 0.98 atm
Therefore, the initial pressure for the flask to be entirely filled with liquid at the end of the process is 0.98 atm.