To solve this problem, we need to apply the principles of gas laws, particularly the ideal gas law, which states that for a given amount of gas, the pressure (P), volume (V), and temperature (T) are related by the equation PV = nRT, where n is the number of moles of gas and R is the ideal gas constant. In this scenario, we have two identical containers connected by a small pipe, and we need to determine the new common pressure when one container is heated while the other remains at a constant temperature.
Initial Conditions
Let's denote the initial pressure in both containers as 'p' and the initial temperature as 'T'. Since both containers are identical and contain the same gas, they will have the same volume, which we can denote as 'V'. Therefore, the initial state of the gas in both containers can be described as:
- Container 1: Pressure = p, Temperature = T
- Container 2: Pressure = p, Temperature = T
Heating One Container
Now, we heat Container 2 to a temperature of 2T while keeping Container 1 at temperature T. According to the ideal gas law, the pressure of a gas is directly proportional to its temperature when the volume is constant. Therefore, we can express the pressures in both containers after heating:
- For Container 1 (at temperature T):
- P1 = (nRT) / V = p
- For Container 2 (at temperature 2T):
- P2 = (nR(2T)) / V = 2p
Finding the Common Pressure
After heating, the gas in Container 2 will exert a pressure of 2p. However, since the two containers are connected, the gases will mix through the small pipe until they reach a common pressure. To find this common pressure, we can use the principle of conservation of energy and the fact that the total number of moles of gas remains constant.
Let’s denote the total volume of the system as 2V (since there are two containers of volume V each). The total number of moles of gas in both containers can be expressed as:
- n1 = (pV) / (RT)
- n2 = (2pV) / (R(2T)) = (pV) / (RT)
Thus, the total number of moles of gas in the system is:
n_total = n1 + n2 = (pV) / (RT) + (pV) / (RT) = 2(pV) / (RT)
Calculating the Final Pressure
Now, we can find the common pressure (P_common) in the system using the ideal gas law again:
P_common * (2V) = n_total * R * T_avg
Where T_avg is the average temperature of the gases in the two containers:
T_avg = (T + 2T) / 2 = (3T) / 2
Substituting the values into the equation:
P_common * (2V) = (2(pV) / (RT)) * R * (3T) / 2
P_common * (2V) = 3pV
Now, dividing both sides by 2V:
P_common = (3p) / 2
Final Answer
Thus, the common pressure of the gases in the two containers after one has been heated to 2T is:
P_common = (3/2)p
Therefore, the correct answer is option a: 3/2 p.
