Question

What is the molar volume of an ideal gas at \(1.00\) atm and (a) \(212^{\circ} \mathrm{F}\); (b) at the normal sublimation point of dry ice \(\left(-78.5^{\circ} \mathrm{C}\right)\) ?

Short answer

The molar volume of an ideal gas is 30.6 L/mol at 212°F and 1.00 atm, and 15.98 L/mol at the normal sublimation point of dry ice (-78.5°C) and 1.00 atm.

Step-by-step solution

Convert Fahrenheit to Kelvin

Convert the temperature from degrees Fahrenheit to Kelvin using the formula: K = (F - 32) * 5/9 + 273.15. For 212°F, the conversion is: K = (212 - 32) * 5/9 + 273.15 = 373.15 K.

Convert Celsius to Kelvin

Convert the temperature reading from degrees Celsius to Kelvin using the formula: K = C + 273.15. For -78.5°C the conversion is: K = -78.5 + 273.15 = 194.65 K.

Apply the Ideal Gas Law

Use the ideal gas law equation, PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the ideal gas constant (0.0821 L atm/mol K), and T is temperature in Kelvin. Since we are looking for molar volume (V/n), we can rearrange the equation to V/n = RT/P.

Calculate Molar Volume for 212°F

Plug in the values of R, T, and P into the V/n equation for the temperature at 212°F (373.15 K): V/n = (0.0821 L atm/mol K)(373.15 K)/(1.00 atm) = 30.6 L/mol.

Calculate Molar Volume for -78.5°C

Repeat the calculation with the temperature at -78.5°C (194.65 K): V/n = (0.0821 L atm/mol K)(194.65 K)/(1.00 atm) = 15.98 L/mol.

Key concepts

Ideal Gas Law

The ideal gas law is an equation of state for a hypothetical gas called an 'ideal gas.' Ideal gases follow the equation PV = nRT, where P is the pressure, V is the volume, n is the number of moles of the gas, R is the gas constant, and T is the absolute temperature in kelvins.
This relationship assumes that the gas particles do not attract or repel each other and have no volume of their own; real gases, of course, do not meet these criteria, but the ideal gas law serves as a useful approximation under many conditions. The law lets us predict the behavior of a gas when conditions such as pressure or temperature change. It's particularly useful for understanding the molar volume, which is the volume one mole of gas occupies under certain conditions of temperature and pressure.
In the context of practical exercises, like the calculation of the molar volume of an ideal gas at a given temperature and pressure, the ideal gas law can be rearranged to solve for the variable of interest. In this case, to find the molar volume (V/n), one would rearrange the equation to V/n = RT/P.

Temperature Conversions

Temperature conversions are crucial when working with the ideal gas law because the temperature must be in kelvins (K). There's a straightforward conversion for Celsius to Kelvin: you simply add 273.15 to the Celsius temperature. Fahrenheit to Kelvin, however, requires a two-step process: first, we subtract 32 from the Fahrenheit temperature, then multiply the result by 5/9, and finally, add 273.15 to get the temperature in Kelvin.

Why Kelvin?

The Kelvin scale is used because it's an absolute temperature scale based on absolute zero, the point at which particles theoretically stop moving (hence, no lower temperature exists). This is unlike the Celsius or Fahrenheit scales, where the numbers are relative to the freezing and boiling points of water (which are not absolute reference points). In scientific calculations, using Kelvin helps to avoid negative numbers and provides consistency, especially when using the ideal gas law.

Molar Volume Calculations

Molar volume is the volume occupied by one mole of a substance. For gases, it varies depending on the temperature and pressure. To calculate the molar volume of an ideal gas, we use the modified form of the ideal gas law: V/n = RT/P.
This formula reveals that the molar volume of an ideal gas is directly proportional to its temperature (T) and inversely proportional to its pressure (P), with the ideal gas constant (R) as a factor. By inputting the measured temperature in Kelvin and known pressure, along with the constant R (which has a value of approximately 0.0821 L atm/mol K for ideal gases), one can compute the molar volume.

Applying the Concept

For instance, at standard temperature and pressure (0°C and 1 atm), one mole of an ideal gas occupies 22.4 liters. However, as the exercise illustrates, at higher temperatures such as 212°F (or 373.15 K when converted), the molar volume increases due to the expanded gas particles, resulting in values like 30.6 L/mol. Conversely, at -78.5°C (or 194.65 K), the molar volume is smaller, at 15.98 L/mol. These calculations are vital in many scientific fields, including chemistry and physics, allowing for predictions on how gases will behave under varying conditions.