To tackle this problem, we need to analyze the behavior of an ideal gas in a cylindrical tube with a diathermic separator. The key points to consider are the properties of the gas, the nature of the separator, and the implications of the adiabatic walls. Let's break it down step by step.
Understanding the System
We have a cylindrical tube with adiabatic walls, meaning no heat can enter or leave the system. The diathermic separator allows heat transfer but does not impede the flow of gas. Initially, the gas is injected into both sides of the tube at equal pressures and temperatures, keeping the separator in the middle. When the separator is moved to divide the tube in a 1:3 ratio, we need to determine the new pressure ratio in the two sections.
Applying the Ideal Gas Law
The ideal gas law states that:
- P = Pressure
- V = Volume
- n = Number of moles of gas
- R = Ideal gas constant
- T = Temperature
In our case, since the temperature remains constant and the gas is ideal, we can express the relationship between pressure and volume as:
P1 * V1 = P2 * V2
Volume Considerations
Let’s denote the total volume of the tube as V. When the separator is moved to create a 1:3 ratio, we can define:
- V1 = Volume of the first section = V/4
- V2 = Volume of the second section = 3V/4
Pressure Ratio Calculation
Since the gas is injected at equal pressures initially, we can denote the initial pressure in both sections as P0. After moving the separator, we can denote the pressures in the two sections as P1 and P2. Using the ideal gas law, we can set up the following equations:
P1 * (V/4) = n1 * R * T
P2 * (3V/4) = n2 * R * T
Since the number of moles of gas is proportional to the pressure and volume, we can express the ratio of pressures:
P1 / P2 = (n1 * R * T) / (n2 * R * T) = n1 / n2
Finding the Moles of Gas
Given that the separator divides the tube in a 1:3 ratio, the number of moles of gas in each section will also reflect this ratio. Thus, we can say:
- n1 = k (for some constant k)
- n2 = 3k
Final Pressure Ratio
Substituting these values into our pressure ratio equation gives us:
P1 / P2 = n1 / n2 = k / 3k = 1 / 3
Rearranging this, we find:
P1 : P2 = 1 : 3
Thus, the ratio of the pressures in the two parts of the vessel is indeed 3:1, confirming the answer.
Summary
In summary, by applying the ideal gas law and considering the volumes and moles of gas in each section after moving the separator, we determined that the pressure ratio in the two parts of the vessel is 3:1. This illustrates how the distribution of gas and the properties of the system interact to maintain equilibrium under the given conditions.
