What is the final pressure when two propane tanks of different volumes and initial pressures come to a uniform state at a certain temperature?
The final pressure can be calculated using the ideal gas law formula, which relates the pressure, volume, and temperature of a gas. In this scenario, we have two propane tanks with different initial conditions that are connected to each other and reach a uniform state at a specific temperature.
Let's break down the data given:
- The first propane tank has a volume of 1 m³, pressure of 150 kPa, and temperature of 300 K.
- The second propane tank has a volume of 0.5 m³, pressure of 250 kPa, and temperature of 380 K.
- The two tanks come to a uniform state at 320 K.
To find the final pressure, we can use the ideal gas law equation: P1V1/T1 = P2V2/T2, where P represents pressure, V represents volume, and T represents temperature.
Given:
P1 = 150 kPa
V1 = 1 m³
T1 = 300 K
P2 = 250 kPa
V2 = 0.5 m³
T2 = 380 K
Tf = 320 K (final temperature when the two tanks come to a uniform state)
We can rearrange the formula to solve for the final pressure (Pf):
Pf = (P1*V1*Tf) / (V2*T1)
Now, let's substitute the values into the formula:
Pf = (150 kPa * 1 m³ * 320 K) / (0.5 m³ * 300 K)
Pf = 48000 kPa*m³ / 150 K*m³
Pf = 320 kPa
Therefore, the final pressure when the two propane tanks come to a uniform state at 320 K is 320 kPa.
Understanding Ideal Gas Law:
The ideal gas law is a fundamental equation in the study of gases, which combines the principles of Boyle's Law, Charles's Law, and Avogadro's Law. It describes the behavior of ideal gases under varying conditions of pressure, volume, and temperature. The ideal gas law equation is expressed as:
PV = nRT
Where:
- P is the pressure of the gas
- V is the volume of the gas
- n is the number of moles of gas
- R is the ideal gas constant
- T is the temperature of the gas in Kelvin
When applying the ideal gas law to solve problems involving gases, it is crucial to ensure that the units of pressure, volume, and temperature are in the appropriate form (usually kPa, m³, and K, respectively) to maintain consistency in calculations.
Significance of Gas Pressure:
Pressure is a critical property of gases that reflects the force exerted by gas molecules on the walls of their container. In the context of propane tanks or any gas container, understanding pressure is essential for ensuring safe and efficient storage and utilization of gases. Monitoring and controlling gas pressure help prevent leaks, regulate flow rates, and maintain the stability of gas systems.
In the given scenario of propane tanks coming to a uniform state, the final pressure serves as a key indicator of how the gases have equilibrated and distributed their properties between the two tanks. By applying the ideal gas law, we can accurately determine the final pressure and predict the behavior of gases in interconnected systems.
In conclusion, the calculation of the final pressure when two propane tanks with different initial conditions come to a uniform state at a specific temperature exemplifies the practical application of the ideal gas law in solving real-world gas problems.