A cylinder with a valve at the bottom is filled with an ideal gas. The valve is now opened and some of the gas escapes slowly. The valve is then closed, after which the piston is observed to be at a lower position. Assume that the system is in thermal equilibrium with the surroundings at all times.
a) Is the final pressure of the gas in the cylinder greater than, less than, or equal to the initial pressure? Explain.
Explain how your answer is consistent with the forces acting on the piston in the initial and final states.
b) In this process, which of the quantities P, V, n, and T are held constant and which are allowed to change?
c) Consider the following incorrect student statement.
"In the ideal gas law, P=nRT/V, so the pressure is inversely proportional to the volume. If you decrease the volume, the pressure has to go up."
What is the flaw in the student's reasoning?
d) Explain why it is not possible to use the ideal gas law to determine whether the pressure changed in this process.
a) The final pressure of the gas in the cylinder will be less than the initial pressure. This can be explained by considering the forces acting on the piston in the initial and final states.
Initially, when the valve is closed, the gas inside the cylinder is under pressure. This creates a force on the piston pushing it upward. The opposing force is the atmospheric pressure acting on the top surface of the piston.
When the valve is opened, some of the gas escapes slowly. As the gas escapes, the number of gas molecules inside the cylinder decreases, resulting in a decrease in the overall force exerted by the gas on the piston. However, the atmospheric pressure acting on the top surface of the piston remains constant.
With a decrease in the force exerted by the gas on the piston and a constant force opposing it, the net force on the piston decreases. This causes the piston to move to a lower position, indicating a decrease in pressure.
b) In this process:
- The pressure (P) is allowed to change.
- The volume (V) is allowed to change as the gas escapes through the valve.
- The number of molecules (n) is not explicitly mentioned in the given scenario, so it is unclear whether it is held constant or allowed to change.
- The temperature (T) is held constant as the system is in thermal equilibrium with the surroundings.
c) The flaw in the student's reasoning lies in assuming a simple inverse relationship between pressure and volume based on the ideal gas law, P = nRT/V. While this equation is correct, it does not imply a direct proportionality between pressure and volume. The equation shows that, assuming all other variables are constant, an decrease in volume (V) would lead to an increase in pressure (P). However, this assumption does not account for changes in any of the other variables.
In reality, the pressure and volume of a gas are interdependent variables that can change in complex ways depending on the specific conditions of the system. Therefore, it is not accurate to conclude that decreasing volume always leads to an increase in pressure.
d) The ideal gas law, P = nRT/V, relates pressure (P), volume (V), number of moles of gas (n), and temperature (T) for an ideal gas. However, in this given scenario, the question only provides information about the changes in pressure and volume, while the number of moles of gas and temperature are not given or mentioned.
To determine whether the pressure changed in this process using the ideal gas law, we would need to know either the initial or final values of either the number of moles of gas (n) or the temperature (T), in addition to the initial and final volume (V). Without this information, it is not possible to use the ideal gas law to directly determine whether the pressure changed.