A glass bulb with radius 15 cm is filled with water vapor at a temperature of 500 K. On the exterior of the bulb a spot with area 2.0 mm^2 is placed in contact with liquid nitrogen. This causes the temperature of the spot to drop well below 0◦C. When the gaseous water molecules strike this area from inside they immediately freeze effectively removing them from the gas. Determine how much time it takes for the pressure in the bulb to reduce to half its original value. Assume the temperature of the gas in the container remains at 500 K
To determine the time it takes for the pressure in the bulb to reduce to half its original value, we need to consider the process of freezing and the resulting change in pressure.
The first step is to calculate the initial pressure inside the bulb. We can use the ideal gas law, which states that PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the gas constant, and T is the temperature in Kelvin.
Since the problem states that the temperature of the gas remains constant at 500 K, we can take this as the initial temperature. The volume of the bulb can be calculated using the formula for the volume of a sphere: V = (4/3) * π * r^3, where r is the radius. Plugging in the values, we find the initial volume.
Next, we need to consider the freezing process. As the gaseous water molecules strike the spot in contact with liquid nitrogen, they freeze and are effectively removed from the gas. This means the number of moles of water vapor decreases over time.
Let's assume that the freezing process is rapid enough that the number of water vapor molecules effectively reduces to zero in a very short period. This means the only remaining gas in the bulb is the remaining gas molecules, which we'll call it M, and the total number of moles is now just M.
As the number of moles decreases, the pressure will also decrease. We can calculate the final pressure using the ideal gas law, where n is replaced with M (the remaining moles) and V remains the same.
The problem states that we want to find the time it takes for the pressure to reduce to half its original value. So we need to find the time it takes for the pressure to decrease from P (the initial pressure) to P/2 (half its original value).
To do this, we can use Boyle's law, which states that P1V1 = P2V2, where P1 is the initial pressure, V1 is the initial volume, P2 is the final pressure, and V2 is the final volume (which remains the same).
Rearranging the equation, we can solve for M, the remaining moles of gas in the bulb.
Finally, we can calculate the time it takes for the pressure to reduce to half its original value using the equation for the rate of change of moles with respect to time: dM/dt = -k * M, where k is a constant.
Integrating this differential equation will give us the time it takes for the pressure to decrease to half its original value.