Question

A cylinder of oxygen gas contains \(91.3 \mathrm{~g} \mathrm{O}_{2}\). If the volume of the cylinder is \(8.58 \mathrm{~L}\), what is the pressure of the \(\mathrm{O}_{2}\) if the gas temperature is \(21^{\circ} \mathrm{C} ?\)

Short answer

The pressure of the oxygen gas in the cylinder is approximately 8.01 atm.

Step-by-step solution

Convert Celsius to Kelvin

First, convert the given temperature from degrees Celsius to Kelvin. This is done using the formula: \( T(K) = T(°C) + 273.15 \). For the given temperature \( 21^{\circ} \text{C} \), the conversion is \( 21 + 273.15 = 294.15 \text{ K} \).

Calculate Moles of Oxygen

Use the molar mass of oxygen to convert grams to moles. The molar mass of \( \mathrm{O}_2 \) is approximately \( 32.00 \text{ g/mol} \). Calculate the moles using the formula: \( \ = \frac{\text{mass}}{\text{molar mass}} \). Thus, \( n = \frac{91.3 \text{ g}}{32.00 \text{ g/mol}} \approx 2.853 \text{ mol} \).

Use Ideal Gas Law to Find Pressure

Apply the ideal gas law equation \( PV = nRT \) where: \( P \) is pressure, \( V \) is volume, \( n \) is moles, \( R \) is the gas constant (0.0821 L·atm/mol·K), and \( T \) is temperature. Solve for \( P \):\[ P = \frac{nRT}{V} \]Substitute the values: \( P = \frac{2.853 \text{ mol} \times 0.0821 \text{ L·atm/mol·K} \times 294.15 \text{ K}}{8.58 \text{ L}} \approx 8.01 \text{ atm} \).

Key concepts

Moles of Gas

When dealing with gases, knowing the number of moles is fundamental to understanding various properties, such as pressure and volume. Moles represent the amount of substance and help in converting grams into a usable chemical quantity. For oxygen, which has a molar mass of about 32.00 g/mol, determining the moles of gas within a given mass involves a straightforward calculation.

• Use the formula: \[ n = \frac{\text{mass}}{\text{molar mass}} \]
• In our example with 91.3 g of O extsubscript{2}, the calculation becomes:\[ n = \frac{91.3 \text{ g}}{32.00 \text{ g/mol}} \approx 2.853 \text{ mol} \]

Calculating the moles allows us to utilize equations like the ideal gas law where moles are a crucial component. It's essential to know this number to evaluate other properties of the gas accurately.

Temperature Conversion

Temperature conversion is crucial when working with gas calculations since gases behave differently depending on the temperature scale used. In science, Kelvin is the preferred unit because it starts at absolute zero and avoids negative temperatures, simplifying many calculations. To convert Celsius to Kelvin:

• Use the formula: \[ T(K) = T(°C) + 273.15 \]
• For example, at 21°C, the conversion is: \[ 21 + 273.15 = 294.15 \text{ K} \]

By converting to Kelvin, we standardize the temperature input, ensuring we apply the ideal gas law correctly. This conversion helps maintain consistency, which is vital for obtaining accurate results.

Pressure Calculation

The pressure of a gas is a key property describing how forcefully the gas particles collide with the walls of their container. To calculate this using the ideal gas law, integrate several key components:- Pressure (P)- Volume (V)- Moles of gas (n)- Gas constant (R): 0.0821 L·atm/mol·K- Temperature in Kelvin (T)The ideal gas law is expressed as:\[ PV = nRT \]To find pressure, rearrange the equation:\[ P = \frac{nRT}{V} \]Substitute known values into the equation:

• Given:\[ n = 2.853 \text{ mol}, \ R = 0.0821 \text{ L·atm/mol·K}, \ T = 294.15 \text{ K}, \ V = 8.58 \text{ L} \]
• Calculate as:\[ P = \frac{2.853 \text{ mol} \times 0.0821 \text{ L·atm/mol·K} \times 294.15 \text{ K}}{8.58 \text{ L}} \approx 8.01 \text{ atm} \]

Understanding this calculation helps in various scientific applications, providing insights about the state of the gas under specific conditions.