Question

A compressed gas cylinder contains \(1.00 \times 10^{3} \mathrm{g}\) argon gas. The pressure inside the cylinder is \(2050 .\) psi (pounds per square inch) at a temperature of \(18^{\circ} \mathrm{C}\) . How much gas remains in the cylinder if the pressure is decreased to 650 . psi at a temperature of \(26^{\circ} \mathrm{C} ?\)

Short answer

There are approximately 359.14 grams of Argon gas remaining in the cylinder when the pressure is decreased to 650 psi at a temperature of 26°C.

Step-by-step solution

1. Convert mass of Argon to moles

To calculate the moles of Argon, we can use the equation: moles = mass / molar mass. The molar mass of Argon is 39.95 g/mol. Therefore: moles = \( \frac{1.00 \times 10^{3} \mathrm{g}}{39.95 \mathrm{g/mol}} \) moles = 25.03 mol

2. Convert psi to atm

We need to convert pressure from psi to atm, because the ideal gas law uses pressure in atm. The conversion factor is 1 atm = 14.696 psi. Therefore: Initial pressure = \( \frac{2050 \mathrm{psi}}{14.696 \mathrm{psi/atm}} \) = 139.58 atm Final pressure = \( \frac{650 \mathrm{psi}}{14.696 \mathrm{psi/atm}} \) = 44.24 atm

3. Convert Celsius to Kelvin

We need to convert temperature from degrees Celsius to Kelvin, because the ideal gas law uses temperature in Kelvin. To convert, add 273.15 to the Celsius temperature. Therefore: Initial Temperature = 18°C + 273.15 = 291.15 K Final Temperature = 26°C + 273.15 = 299.15 K

4. Calculate the volume of the cylinder

Using the ideal gas law, PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the gas constant (0.0821 \( \frac{\mathrm{L\cdot atm}}{\mathrm{mol\cdot K}} \)), and T is temperature, we can calculate the volume of the cylinder: V = \( \frac{nRT}{P} \) = \( \frac{(25.03 \mathrm{mol})(0.0821 \frac{\mathrm{L\cdot atm}}{\mathrm{mol\cdot K}})(291.15 \mathrm{K})}{139.58 \mathrm{atm}} \) V = 4.86 L

5. Calculate the moles of Argon remaining

We can use the ideal gas law again but with the final pressure and temperature values to determine the moles of Argon remaining in the cylinder: n_final = \( \frac{PV}{RT} \) = \( \frac{(44.24 \mathrm{atm})(4.86 \mathrm{L})}{(0.0821 \frac{\mathrm{L\cdot atm}}{\mathrm{mol\cdot K}})(299.15 \mathrm{K})} \) n_final = 8.99 mol

6. Convert moles of Argon back to grams

Finally, we can convert the moles of Argon remaining back to grams using the molar mass of Argon (39.95 g/mol): Mass remaining = (8.99 mol) (39.95 g/mol) = 359.14 g There are approximately 359.14 grams of Argon gas remaining in the cylinder.

Key concepts

Argon Gas

Argon gas is a noble gas, represented by the symbol Ar on the periodic table. It's one of the most abundant gases in the Earth's atmosphere, making up about 0.93% by volume. Unlike other gases, argon is characterized by its lack of reactivity with most substances, which makes it ideal for use in various applications. In this exercise, we work with argon in a compressed gas cylinder.

• The molecular weight of argon is 39.95 g/mol, meaning in any sample of argon gas, each mole weighs about 39.95 grams.
• Because argon is chemically inert, it is often used in situations where materials need to be protected from reactions. This includes environments with corrosion risks, where it acts as a shielding gas.

Understanding argon's properties helps us calculate how much gas remains in a cylinder, especially when the conditions such as pressure or temperature change.

Pressure Conversion

Pressure conversion is crucial when working with gases, as different units can be used depending on the region or industry standard. In this problem, the initial pressure is given in psi (pounds per square inch), but we need to convert it into atm (atmospheres) to use the ideal gas law equation.

• The conversion factor we use here is 1 atm = 14.696 psi.
• This allows us to equate different pressure readings, converting 2050 psi to 139.58 atm, and 650 psi to 44.24 atm.

It's critical to perform this conversion accurately, as errors could significantly affect the outcome of any calculations involving gas laws. Always make sure to use the correct conversion factor to avoid any inconsistencies.

Temperature Conversion

Temperature conversion is another vital task when working with the ideal gas law. The ideal gas law requires temperature to be in Kelvin, as it links the absolute temperature scale directly to volume and pressure relations in gases.

• To convert Celsius to Kelvin, simply add 273.15 to the Celsius temperature.
• Thus, 18°C becomes 291.15 K, and 26°C becomes 299.15 K.

Kelvin is preferred in scientific calculations because it starts at absolute zero, where all thermal motion ceases. This linear relationship with energy makes the Kelvin scale and appropriate conversions essential for accurate results in thermodynamics and gas law calculations.

Moles Calculation

Calculating moles is a fundamental step when using the ideal gas law, expressed as PV = nRT, where "n" is the number of moles of gas. Moles give us a way to quantify the amount of a substance.

• Initially, the total moles of argon are determined by dividing the given mass (1000 g) by the molar mass (39.95 g/mol), which yields about 25.03 moles.
• To find out how much argon is left after the change in conditions, we use the ideal gas law with the final conditions to find that 8.99 moles remain.

Mole calculations provide precise insight into the quantity of gas, enabling predictions of behavior under different physical conditions. These calculations are instrumental when considering reactions or changes in gas properties over time.