Question

Oxygen is supplied to a medical facility from ten \(1.5-\mathrm{ft}^{3}\) compressed oxygen tanks. Initially, these tanks are at 1500 psia and \(80^{\circ} \mathrm{F}\). The oxygen is removed from these tanks slowly enough that the temperature in the tanks remains at \(80^{\circ} \mathrm{F}\). After two weeks, the pressure in the tanks is 300 psia.

Short answer

Answer: The final quantity of oxygen remaining in the ten tanks is \(m_{final}\) kg (calculated using the step-by-step solution provided).

Step-by-step solution

Use Ideal Gas Law formula for initial and final states

The Ideal Gas Law is given by the formula: PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the universal gas constant and T is temperature in Kelvin. We must use the Ideal Gas Law for both initial and final states: Initial state: \(P_1 = 1500 \,\text{psia}\), \(V_1 = 1.5\,\text{ft}^{3}\), \(T_1 = 80^{\circ} \mathrm{F}\) Final state: \(P_2 = 300\,\text{psia}\), \(V_2 = 1.5\,\text{ft}^{3}\), \(T_2 = 80^{\circ} \mathrm{F}\) Remember to convert Fahrenheit to Kelvin: \(T_{1(K)} = (T_1 + 459.67) * \frac{5}{9}\), \(T_{2(K)} = (T_2 + 459.67) * \frac{5}{9}\) Also, convert psia (pound per square inch absolute) to Pa (pascal) with the appropriate conversion factor: \(P_{1(Pa)} = P_1 * 6895\), \(P_{2(Pa)} = P_2 * 6895\).

Calculate the initial and final number of moles of oxygen in one tank

Using the Ideal Gas Law for initial and final states, calculate the initial and final number of moles of oxygen, \(n_1\) and \(n_2\): \(n_1 = \frac{P_{1(Pa)}V_1}{RT_{1(K)}}\), \(n_2 = \frac{P_{2(Pa)}V_2}{RT_{2(K)}}\) Remember to convert cubic feet to cubic meters: \(V_{1(m^3)} = V_1 * 0.0283168\).

Calculate the initial and final mass of oxygen in one tank

Given the number of moles, you can find the mass, knowing the molar mass of oxygen: \(M_{O_2} = 32\,g/mol\). Convert it to \(kg/mol\): \(M_{O_2(kg/mol)} = 0.032\,kg/mol\). Then calculate the initial and final mass of oxygen in one tank: \(m_1 = n_1 * M_{O_2(kg/mol)}\), \(m_2 = n_2 * M_{O_2(kg/mol)}\)

Calculate the difference in mass and the final quantity of oxygen in all ten tanks

Calculate the mass of oxygen used in one tank: \(\Delta m = m_1 - m_2\). Multiply this value by 10 to get the mass of oxygen used in all ten tanks: \(\Delta m_{total} = 10 \cdot \Delta m\). Then, subtract this value from the initial mass in all tanks to find the final quantity of oxygen: \(m_{final} = 10m_1 - \Delta m_{total}\).