An oxygen tank of volume V is stored in a medical laboratory. Initially, the tank is full of compressed oxygen at pressure p_{0} . Lab regulations require a refill when the pressure in the tank drops to p_{2} so as to avoid contaimination by impure gas. Suppose that the daily consumption of oxygen out of the tank is a volume V_{0} at the standard pressure p_{0} and that the temperature T of the laboratory remains constant throughout all processes. Treat the oxygen as an ideal gas and find how many days can this tank of oxygen be used before the first refill becomes necessary. [Hint: Keep track of the mass of oxygen using the equation of state for an ideal gas]
To find the number of days before the first refill becomes necessary, we need to determine the amount of oxygen consumed each day and compare it to the total volume of the tank.
Let's start by understanding the properties of the gas using the ideal gas law:
PV = nRT
where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant, and T is the temperature.
In this case, we want to find the number of days, so let's express time in terms of days and set up our equation:
(P0 * V0) * (1 day) = nRT
To relate the volume of the tank to the number of days, we need to calculate the total number of moles of oxygen in the tank initially and subtract the amount consumed each day.
Calculating the initial number of moles:
n_initial = (P0 * V) / (RT)
Next, let's calculate the number of moles consumed each day:
n_consumed = (P0 * V0) / (RT)
To find the number of days before refill, we need to find how many days it takes for the initial number of moles to be consumed to the refill pressure:
n_initial - (n_consumed * d) = (P2 * V) / (RT)
where d is the number of days.
Rearranging the equation to solve for the number of days:
d = [(n_initial - (P2 * V) / (RT)] / n_consumed
Now, we have all the necessary information to solve the problem. Plug in the values for P0, V, V0, T, and P2 into the equation, and calculate the number of days.
To solve this problem, we need to use the ideal gas law equation:
PV = nRT
where P is the pressure, V is the volume, n is the number of moles of gas, R is the ideal gas constant, and T is the temperature.
We can rearrange this equation to solve for n:
n = PV / RT
Since the temperature is constant, we can simplify the equation further:
n = (P / RT) * V
Now, let's consider the daily consumption of oxygen. We are given that the daily consumption is a volume V0 at the standard pressure P0. Using the equation of state, we can calculate the number of moles of oxygen consumed each day:
n0 = (P0 / RT) * V0
To find out how many days the tank can be used before the refill is necessary, we need to determine the number of moles of oxygen in the tank initially, and then calculate how long it takes to decrease to a pressure of P2.
The initial number of moles of oxygen in the tank is given by:
n_initial = (P0 / RT) * V
Now, we need to find the number of moles of oxygen remaining in the tank when the pressure drops to P2. Using the ideal gas law again:
P2V = n_remaining * RT
Simplifying for n_remaining:
n_remaining = (P2 / RT) * V
The difference in the number of moles of oxygen is given by:
Δn = n_initial - n_remaining
Since the daily consumption is n0, we can calculate the number of days before the first refill becomes necessary:
Number of days = Δn / n0
Substituting the equations:
Number of days = (n_initial - n_remaining) / n0
Number of days = ( (P0 / RT) * V - (P2 / RT) * V ) / (P0 / RT) * V0
Simplifying:
Number of days = (P0 - P2) / P0 * V / V0
So, the number of days that the tank of oxygen can be used before the first refill becomes necessary is (P0 - P2) / P0 * V / V0.