Question

A \(13-\mathrm{m}^{3}\) tank contains nitrogen at \(17^{\circ} \mathrm{C}\) and \(600 \mathrm{kPa}\) Some nitrogen is allowed to escape until the pressure in the tank drops to \(400 \mathrm{kPa}\). If the temperature at this point is \(15^{\circ} \mathrm{C},\) determine the amount of nitrogen that has escaped.

Short answer

Answer: Approximately 1271.3 moles of nitrogen have escaped from the tank.

Step-by-step solution

Convert temperature and pressure to Kelvin and Pascals

First, we need to convert the given temperatures and pressures to Kelvin (K) and Pascals (Pa), respectively, to use with the ideal gas law. To do this, we use the following formulas: - Convert Celsius to Kelvin: \(T_K = T_C + 273.15\) - Convert kilopascals to Pascals: \(P_{Pa} = P_{kPa} \times 1000\) Initial state: - Temperature: \(T1 = (17+273.15) \ \mathrm{K} = 290.15 \ \mathrm{K}\) - Pressure: \(P1 = (600) \times 1000 \ \mathrm{Pa} = 6\times10^{5} \ \mathrm{Pa}\) Final state: - Temperature: \(T2 = (15+273.15) \ \mathrm{K} = 288.15 \ \mathrm{K}\) - Pressure: \(P2 = (400) \times 1000 \ \mathrm{Pa} = 4\times10^{5} \ \mathrm{Pa}\)

Calculate the initial and final moles of nitrogen in the tank using the ideal gas law

The ideal gas law is \(PV = nRT\), where \(P\) is pressure, \(V\) is volume, \(n\) is the number of moles, \(R\) is the ideal gas constant, and \(T\) is temperature. Rearrange the equation to solve for \(n\): \(n = \frac{PV}{RT}\) Using the ideal gas constant for nitrogen, \(R = 8.314 \ \mathrm{J/(mol \cdot K)}\), we can calculate the number of moles for the initial and final states: Initial moles: \(n1 = \frac{P1 \cdot V}{R \cdot T1} = \frac{(6\times10^{5} \ \mathrm{Pa})(13\ \mathrm{m^3})}{(8.314 \ \mathrm{J/(mol \cdot K)})(290.15 \ \mathrm{K})} = 3207.6\ \mathrm{mol}\) Final moles: \(n2 = \frac{P2 \cdot V}{R \cdot T2} = \frac{(4\times10^{5} \ \mathrm{Pa})(13\ \mathrm{m^3})}{(8.314 \ \mathrm{J/(mol \cdot K)})(288.15 \ \mathrm{K})} = 1936.3\ \mathrm{mol}\)

Determine the amount of nitrogen that has escaped

To find out the amount of nitrogen that has escaped, subtract the final moles from the initial moles: \(n_{escaped} = n1 - n2 = 3207.6 \ \mathrm{mol} - 1936.3 \ \mathrm{mol} = 1271.3 \ \mathrm{mol}\) The amount of nitrogen that has escaped is approximately 1271.3 moles.