Question

A compressed gas cylinder contains \(1.00 \times 10^{3} \mathrm{~g}\) argon gas. The pressure inside the cylinder is \(2050 . \mathrm{psi}\) (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

The amount of argon gas that remains in the cylinder after the pressure and temperature change is approximately \(310 \mathrm{g}\).

Step-by-step solution

Convert the initial pressure to atm

First, we need to convert the initial pressure from pounds per square inch (psi) to atmospheres (atm). To do this, we can use the following conversion factor: 1 psi = 0.068046 atm The initial pressure in atm is: \(P_1 = 2050 \text{ psi} \times 0.068046 \frac{\text{atm}}{\text{psi}} = 139.4943 \text{ atm}\)

Convert the final pressure to atm

Next, we convert the final pressure from psi to atm: \(P_2 = 650 \text{ psi} \times 0.068046 \frac{\text{atm}}{\text{psi}} = 44.2299 \text{ atm}\)

Convert the initial and final temperatures to Kelvin

To convert the initial temperature from Celsius to Kelvin, add 273.15: \(T_1 = 18^{\circ}\mathrm{C} + 273.15 = 291.15 \mathrm{K}\) Similarly, for the final temperature: \(T_2 = 26^{\circ}\mathrm{C} + 273.15 = 299.15 \mathrm{K}\)

Convert the amount of gas to moles

The initial amount of argon gas given in grams need to be converted to moles. The molar mass of argon is approximately 39.948 g/mol. Calculate the initial moles of argon gas: \(n_1 = \frac{1.00 \times 10^3 \mathrm{g}}{39.948 \frac{\mathrm{g}}{\mathrm{mol}}} = 25.06469 \mathrm{mol}\)

Calculate the final amount of gas in moles

Compare the ratio in the ideal gas law: \(\frac{n_1T_1}{P_1} = \frac{n_2T_2}{P_2}\) Then, solve for the final amount of gas in moles: \(n_2 = \frac{n_1T_1}{P_1} \times \frac{P_2}{T_2} = \frac{25.06469 \mathrm{mol} \times 291.15 \mathrm{K}}{139.4943 \mathrm{atm}} \times \frac{44.2299 \mathrm{atm}}{299.15 \mathrm{K}} = 7.763325 \mathrm{mol}\)

Convert the final amount of gas to grams

Finally, convert the final amount of argon gas in moles back to grams: Final amount of argon gas remaining: \(7.763325 \mathrm{mol} \times 39.948 \frac{\mathrm{g}}{\mathrm{mol}} = 310.0186 \mathrm{g}\) The amount of argon gas that remains in the cylinder after the pressure and temperature change is approximately 310 g.

Key concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation that relates the pressure, volume, temperature, and amount of gas in moles. This law assumes that gas particles are point-like with no volume and do not interact with each other, which is an idealized scenario. The law is represented by the equation
\[ PV = nRT \]
where

• \(P\) is the pressure of the gas,
• \(V\) is the volume of the gas,
• \(n\) is the number of moles of the gas,
• \(R\) is the universal gas constant, and
• \(T\) is the temperature of the gas in Kelvin.

In the context of the given exercise, the Ideal Gas Law allows us to find the relationship between the state of argon gas at two different sets of conditions through the combined gas law formula, specifically adjusting the ratio of \(n_1T_1/P_1 = n_2T_2/P_2\). This principle is crucial in predicting how the amount of gas (moles) will change when pressure and temperature conditions vary, provided that the gas behaves ideally.

Converting Units in Chemistry

Unit conversion in chemistry is an essential skill because measurements of physical quantities such as pressure, volume, and temperature can be recorded in various units. To solve chemical problems accurately, these measurements need to be in consistent units.
The pressure unit conversion from psi to atm and the temperature conversion from Celsius to Kelvin in the exercise are examples of such conversions. The pressure was converted by multiplying by 0.068046 atm/psi, which is the proportion between the two units. To convert temperatures from Celsius to Kelvin, 273.15 is added to the Celsius temperature. It's crucial to note that the universal gas constant (\(R\)) has different values depending on the units of pressure and volume used, which is why consistent units are mandatory for calculations involving the ideal gas law.

Moles and Molar Mass

Moles and molar mass are core concepts in stoichiometry, which is the quantitative aspect of chemistry. A mole (symbol: mol) represents a specific number of particles, usually atoms or molecules--Avogadro's number, which is \(6.022 \times 10^{23}\).
The molar mass of a substance is the mass in grams of one mole of that substance. It's the bridge between the mass of a sample and the number of moles because it tells us how much one mole of a substance weighs. In the exercise, the molar mass of argon gas (approximately 39.948 g/mol) is used to convert the mass of argon gas to moles. By dividing the given mass in grams by the molar mass, we obtain the number of moles of argon, which is essential for using the ideal gas law equation.