Question

A piece of dry ice (solid carbon dioxide) with a mass of 5.50 \(\mathrm{g}\) is placed in a 10.0 -L vessel that already contains air at 705 torr and \(24^{\circ} \mathrm{C}\) . After the carbon dioxide has totally sublimed, what is the partial pressure of the resultant CO\(_{2}\) gas, and the total pressure in the container at \(24^{\circ} \mathrm{C} ?\)

Short answer

The partial pressure of CO2 in the container after the dry ice has completely sublimed is approximately \(0.3075\:atm\) and the total pressure is approximately \(1.2351\:atm\) at \(24^{\circ}C\).

Step-by-step solution

Find moles of CO2 produced

First, we need to find the moles of solid CO2 that will sublimate to form gas. Use the mass and molar mass of CO2 (44.01 g/mol) to perform this calculation. moles of CO2 = mass of CO2 / molar mass of CO2 moles of CO2 = 5.50 g / 44.01 g/mol

Calculate the partial pressure of CO2

Next, we will use the ideal gas law to find the partial pressure of CO2. The ideal gas law formula is: PV = nRT Where: P = pressure V = volume n = moles of gas R = ideal gas constant (0.0821 L atm/mol K) T = temperature in Kelvin First, we need to convert the temperature from Celsius to Kelvin. T(K) = T(°C) + 273.15 T(K) = 24 + 273.15 = 297.15 K Now we can use the ideal gas law to find the partial pressure of CO2. P(CO2) = n(CO2)RT / V P(CO2) = (moles of CO2) (0.0821 L atm/mol K) (297.15 K) / 10.0 L

Convert initial pressure to atmospheres

The initial pressure in the container is given in torr. We need to convert this pressure to atmospheres, since our ideal gas constant uses units of "atm." To do so, use the conversion factor: 1 atm = 760 torr Initial pressure (atm) = 705 torr / 760 torr/atm

Calculate the total pressure

Now we can find the total pressure in the container after the CO2 has sublimated. Simply add the partial pressure of CO2 to the initial pressure in the container. Total pressure = Initial pressure (atm) + P(CO2) Final step: fill-in calculations from previous steps moles of CO2 = 5.50 g / 44.01 g/mol = 0.125 mol P(CO2) = (0.125 mol) (0.0821 L atm/mol K) (297.15 K) / 10.0 L = 0.3075 atm Initial pressure (atm) = 705 torr / 760 torr/atm = 0.9276 atm Total pressure = 0.9276 atm + 0.3075 atm = 1.2351 atm The partial pressure of CO2 in the container after the dry ice has completely sublimed is approximately 0.3075 atm and the total pressure is approximately 1.2351 atm at 24°C.

Key concepts

Ideal Gas Law

Understanding ideal gas law is crucial when dealing with problems related to the behavior of gases. The ideal gas law is a fundamental equation that relates the four properties of a gas: pressure (P), volume (V), number of moles (n), and temperature (T). It is commonly expressed as PV = nRT, where R is the ideal gas constant, which can vary depending on the units of pressure, volume, and temperature used in the equation.

For students working with the ideal gas law, remember to pay attention to the units for each value. Pressure is often measured in atmospheres (atm) or torr; volume in liters (L); moles as the amount of substance; and temperature in Kelvin (K). Converting all the units to match (like torr to atm for pressure, or degrees Celsius to Kelvin for temperature) before starting any calculations is paramount for an accurate result.

Sublimation of Dry Ice

Sublimation is a fascinating process where a substance transitions directly from a solid to a gas without passing through the liquid state. Dry ice, which is solid carbon dioxide (CO2), is a common example of a substance that undergoes sublimation. When dry ice is exposed to air at room temperature, it sublimates and forms CO2 gas.

For students observing this phenomenon in an experiment, it's essential to note that the rate of sublimation depends on factors like temperature and air pressure. Your exercise may involve calculating the pressure exerted by the CO2 gas after sublimation—this is where understanding the ideal gas law comes in handy. If you're working on homework, ensure you know the initial mass of the dry ice to calculate the moles of CO2 gas produced, which are then used in the ideal gas law equation.

Molar Mass

Molar mass is the weight of one mole of a substance, typically measured in grams per mole (g/mol). This property is crucial for converting between the mass of a substance and the number of moles present—an indispensable step in stoichiometry and gas law calculations. To find the moles from a given mass, you would use the equation moles = mass / molar mass.

In practice, identifying the molar mass of CO2 (around 44.01 g/mol) allows for the calculation of moles of gas, which is essential when applying the ideal gas law. Students should be meticulous with their calculations and look up the correct molar mass for the relevant substance to avoid any errors.

Conversion of Temperature Units

Conversion of temperature units is a necessary step in many scientific calculations, especially when dealing with the ideal gas law. Since this law requires temperature to be in Kelvin, any temperatures given in Celsius or Fahrenheit must be converted before use.

For Celsius to Kelvin, the conversion formula is T(K) = T(°C) + 273.15. This step must be performed accurately to ensure the correctness of your calculations. Keep in mind there is no such thing as 'negative' Kelvin, as the Kelvin scale starts at absolute zero—the coldest possible temperature.

Students often confuse temperature conversions, so a good tip is to double-check your converted values before plugging them into equations. Accurate conversion is fundamental to deriving the correct final pressure in gas law problems, just as done in the dry ice exercise.