Question

What volume will 5.6 moles of sulfur hexafluoride (SF \(_{6}\) ) gas occupy if the temperature and pressure of the gas are \(128^{\circ} \mathrm{C}\) and \(9.4 \mathrm{~atm} ?\)

Short answer

The volume that 5.6 moles of sulfur hexafluoride (SF6) will occupy under the given conditions (128 degrees Celsius, 9.4 atm) is approximately 20.72 liters.

Step-by-step solution

Convert temperature to Kelvin

It is known that Kelvin temperature is obtained by adding 273.15 to the Celsius temperature. Then, the temperature is \( 128C + 273.15 = 401.15K \)

Use correct value of R

Since the pressure is given in atmospheres, one must use the ideal gas constant which corresponds to atmospheres for pressure. Thus, \( R = 0.0821 L*atm/(K*mol) \)

Apply ideal gas law

Given n=5.6 mol, P=9.4 atm, R=0.0821 L*atm/(K*mol), T=401.15K, you can substitute these values into the ideal gas equation \( PV = nRT \), which, rearranged for V, becomes \( V = nRT/P \). Substitution of all given data into this equation yields the volume V.

Calculate volume

Substitute the values into the formula \( V = nRT/P \) to calculate the volume: \( V = (5.6 mol * 0.0821 L*atm/(K*mol) * 401.15K) / 9.4 atm \). After performing the calculations, the answer comes out to V = 20.72 L.

Key concepts

Molar Volume of Gas

The molar volume of a gas is a fundamental concept in understanding the behavior of gases under various conditions. It refers to the volume that one mole of any gas occupies when certain conditions, usually standard temperature and pressure (STP), are met. At STP, which is defined as 0 degrees Celsius (273.15 Kelvin) and 1 atmosphere of pressure, one mole of any ideal gas occupies 22.4 L.

For substances like sulfur hexafluoride (SF₆) in our exercise, the molar volume can differ from the standard molar volume at conditions other than STP. Using the ideal gas law, we can find the volume occupied by a given amount of moles at specific non-standard conditions of temperature and pressure by manipulating the formula accordingly.

Gas Law Calculations

To perform gas law calculations, it's essential to use the ideal gas law, which is represented by the equation \( PV = nRT \), where \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the ideal gas constant, and \( T \) is the temperature in Kelvin. In our example with sulfur hexafluoride, the calculation involved finding the volume, so the equation was rearranged to solve for \( V \).

By plugging in the values of pressure, moles, temperature, and the ideal gas constant, one can find the volume. It's crucial to use consistent units throughout the calculation, such as atmospheres for pressure, liters for volume, kelvins for temperature, and moles for the amount of substance.

Converting Celsius to Kelvin

Temperature conversions between Celsius and Kelvin are a basic yet vital step in any gas law calculation. Since the ideal gas law requires the absolute temperature in Kelvin, you need to convert Celsius to Kelvin before substituting into the equation. The conversion is straightforward: simply add 273.15 to the Celsius temperature.

In the provided step-by-step solution, we convert the given temperature of \(128^\circ C\) to Kelvin by adding 273.15, resulting in a temperature of 401.15 K. It is this value that is then used in the ideal gas law equation.

Ideal Gas Constant

The ideal gas constant, \( R \), is a vital part of the ideal gas law equation. Its value links the other variables in the equation—pressure, volume, moles, and temperature. The constant has different values based on the units used for pressure and volume. As in our sulfur hexafluoride example, when the pressure is in atmospheres and volume in liters, \( R \) is 0.0821 L*atm/(K*mol).

This value ensures that the units of pressure and volume are compatible for the calculations. Always verify that the correct value of \( R \) is used based on unit requirements of the specific problem to obtain an accurate calculation of the gas volume, as done in our exercise improvement.