Problem 144 - Chemistry An Atoms First Approach

Question

A steel cylinder contains \(150.0\) moles of argon gas at a temperature of \(25^{\circ} \mathrm{C}\) and a pressure of \(8.93 \mathrm{MPa}\). After some argon has been used, the pressure is \(2.00 \mathrm{MPa}\) at a temperature of \(19^{\circ} \mathrm{C}\). What mass of argon remains in the cylinder?

Step-by-Step Solution

Answer

The mass of argon remaining in the cylinder is approximately \(2536 \text{ g}\).

Step 1: Identify the Ideal Gas Law formula

The Ideal Gas Law formula is given by: PV = nRT Where P is the pressure, V is the volume, n is the number of moles, R is the gas constant, and T is the temperature in Kelvin.

Step 2: Convert the given temperatures to Kelvin

The temperatures given are in Celsius, so we need to convert them to Kelvin: Initial temperature: \(T_1 = 25^{\circ}C + 273.15 = 298.15 K\) Final temperature: \(T_2 = 19^{\circ}C + 273.15 = 292.15 K\)

Step 3: Relate the initial and final conditions using the Ideal Gas Law

Since the volume of the cylinder is constant and we are given \(n_1, P_1, T_1, P_2,\) and \(T_2\), we can relate the initial and final conditions as follows: \(\frac{P_1 V}{T_1} = n_1 R \Rightarrow V = \frac{n_1 RT_1}{P_1}\) Similarly, for the final conditions: \(\frac{P_2 V}{T_2} = n_2 R \Rightarrow V = \frac{n_2 RT_2}{P_2}\) Since the volume is constant, the two expressions for V should be equal: \(\frac{n_1 RT_1}{P_1} = \frac{n_2 RT_2}{P_2}\)

Step 4: Solve for the final number of moles of argon, \(n_2\)

We can now solve for the final number of moles of argon, \(n_2\). The gas constant R will cancel out from both sides: \(n_2 = \frac{n_1 P_2 T_1}{P_1 T_2}\) Substitute the given values and solve for \(n_2\): \(n_2 = \frac{150.0 \text{ moles} \times 2.00 \text{ MPa} \times 298.15 \text{ K}}{8.93 \text{ MPa} \times 292.15 \text{ K}} \approx 63.47 \text{ moles}\)

Step 5: Calculate the mass of argon remaining in the cylinder

Now that we have the final number of moles of argon, we can calculate the mass of argon remaining in the cylinder using the molar mass of argon: Molar mass of argon: \(39.95 \frac{\text{g}}{\text{mole}}\) Mass of argon remaining: \(m_2 = n_2 \times \text{Molar mass of argon}\) \(m_2 = 63.47 \text{ moles} \times 39.95 \frac{\text{g}}{\text{mole}} \approx 2536 \text{g}\) Therefore, the mass of argon remaining in the cylinder is approximately 2536 grams.