Question

Dry ice is solid carbon dioxide. A \(0.050 \mathrm{~g}\) sample of dry ice is placed in an evacuated \(4.6 \mathrm{~L}\) vessel at \(30^{\circ} \mathrm{C}\). Calculate the pressure inside the vessel after all the dry ice has been converted to \(\mathrm{CO}_{2}\) gas. (a) \(6.14\) atm (b) \(0.614\) atm (c) \(0.0614 \mathrm{~atm}\) (d) \(6.14 \times 10^{-3}\) atm

Short answer

Using the Ideal Gas Law, the pressure is calculated to be 0.614 atm.

Step-by-step solution

Convert mass of dry ice to moles

Firstly, calculate the number of moles () of carbon dioxide (CO2) using the molecular weight of CO2 (44.01 g/mol). Use the formula = mass / molar mass. For a sample of 0.050 g CO2, n = 0.050 g / 44.01 g/mol.

Use the Ideal Gas Law

Apply the Ideal Gas Law, PV = nRT, where P is the pressure, V is the volume in Liters, n is the number of moles, R is the Ideal Gas Constant (0.0821 L atm/mol K), and T is the temperature in Kelvin. Convert the temperature from Celsius to Kelvin by adding 273.15 to the Celsius temperature.

Solve for the Pressure

Rearrange the Ideal Gas Law to solve for P: P = (nRT)/V. Substitute in the values for n, R, T (in Kelvin), and V to find the pressure in the container.

Key concepts

Dry Ice to CO2 Conversion

Understanding the process of converting dry ice (solid carbon dioxide) into gaseous CO₂ is a quintessential part of solving many chemistry problems.

Consider dry ice, which at room temperature quickly sublimates—that is, it turns directly from a solid into a gas without passing through a liquid phase. This property is leveraged in various applications from creating fog effects to preserving perishable goods. When dry ice sublimates in a closed vessel, the CO₂ gas needs to be quantified, and this is often where the Ideal Gas Law comes into play.

The weight of dry ice given typically must be converted into moles, as the gas laws require the amount of substance in this unit. The molar mass of CO₂ (44.01 g/mol) becomes a cornerstone value for computations. The moles of CO₂ reflect the quantity of gas that will exert pressure in the vessel after sublimation.

Calculating Pressure in Gas Law

The Ideal Gas Law is a valuable tool for calculating the pressure exerted by gases under varying conditions of temperature and volume. Expressed as PV = nRT, where:

• P represents pressure,
• V stands for volume,
• n is the number of moles of gas,
• R is the gas constant (0.0821 L atm/mol K), and
• T is the temperature in Kelvin.

To solve for the pressure following the sublimation of dry ice, one must rearrange this formula to P = (nRT)/V. It's vital to convert all measurements to the appropriate units: volume in liters, temperature in Kelvin, and pressure in atmospheres (atm). With these conversions, applying the formula yields the pressure of CO₂ in the vessel.

Real-world applications, from calculating the pressure needed to carbonate beverages to determining the conditions for gas storage, make understanding this law a highly useful skill.

Molar Mass Calculation

The molar mass of a substance is the mass of one mole of that substance, typically measured in grams per mole (g/mol). It is a fundamental concept in chemistry because it bridges the gap between the atomic scale and the macroscopic world.

Mathematically, molar mass is calculated by adding the average atomic masses of all the atoms in a single molecule of the substance. For CO₂, this involves weighting the atomic mass of carbon (12.01 g/mol) with two oxygen atoms (each 16.00 g/mol), totaling 44.01 g/mol for CO₂.

This calculation is crucial when dealing with chemical reactions and processes, allowing chemists to measure out precise amounts of a substance. In the context of the given exercise, calculating the molar mass of CO₂ is the first step in determining how many moles of gas form from a known mass of dry ice, subsequently affecting the pressure in a closed system.