Question

A \(20 \mathrm{~g}\) chunk of Dry Ice \(\left(\mathrm{CO}_{2}\right)\) is placed in an "empty" \(0.75\) liter wine bottle and tightly corked. What would be the final pressure in the bottle after all the \(\mathrm{CO}_{2}\) has evaporated and the temperature has reached \(25^{\circ} \mathrm{C}\) ?

Short answer

The final pressure in the wine bottle after all the CO₂ has evaporated and the temperature has reached 25°C is approximately 14.14 atm.

Step-by-step solution

Convert mass of CO₂ into moles

The given mass of CO₂ is 20g. We'll need to convert this into moles using the molar mass of CO₂. Molar mass of CO₂ = 12.01g/mol (for C) + (2 × 16.00g/mol for O) = 44.01g/mol Divide the mass of CO₂ by its molar mass to get the number of moles: n = (20 g CO₂) / (44.01 g/mol) = 0.4545 mol

Convert the temperature to Kelvin

We're given the temperature in Celsius (25°C). Convert this to Kelvin by adding 273.15: T = 25°C + 273.15 = 298.15 K

Calculate the final pressure using the ideal gas law equation

Now we have: - V = 0.75 L (volume of the wine bottle) - n = 0.4545 mol (moles of CO₂ obtained from Step 1) - R = 0.0821 atm·L/mol·K (ideal gas constant) - T = 298.15 K (temperature obtained from Step 2) Use the ideal gas law equation (PV=nRT) to solve for the final pressure (P): P = (n × R × T) / V P = (0.4545 mol × 0.0821 atm·L/mol·K × 298.15 K) / 0.75 L P = 14.135 atm The final pressure in the wine bottle after all the CO₂ has evaporated and the temperature has reached 25°C is approximately 14.14 atm.

Key concepts

Dry Ice Evaporation

When dealing with Dry Ice evaporation, it's crucial to understand that dry ice is the solid form of carbon dioxide (CO₂). Unlike water ice, dry ice doesn't melt into a liquid state but instead sublimates directly from solid to gas at temperatures higher than -78.5°C at standard atmospheric pressure. This process is fascinating because it demonstrates the principles of phase changes and thermodynamics.

Dry ice is commonly used for refrigeration, particularly during shipping, because it keeps items colder for longer periods without the mess of water from melting ice. However, when placed in an enclosed space like our wine bottle scenario, the sublimated CO₂ gas increases the pressure inside the container. As a result, if the bottle is sealed tightly and unable to withstand the pressure, it could potentially rupture.

To safely calculate the final pressure inside the bottle, one must employ the ideal gas law, considering the relationship between pressure, volume, temperature, and the amount of substance in the system.

Molar Mass Calculation

To grasp the concept of molar mass calculation, let's delve into what molar mass signifies. It is the mass of one mole of a substance, usually expressed in grams per mole (g/mol). The molar mass is fundamental in chemistry as it bridges the gap between the mass of a material we can measure and the number of atoms or molecules it represents.

Here's a simpler breakdown of how molar mass is calculated:

• Determine the elemental composition of the compound.
• Find the atomic mass of each element from the periodic table. For carbon (C), it's approximately 12.01 g/mol, and for oxygen (O), it's about 16.00 g/mol.
• Combine the atomic masses based on the compound's formula. For CO₂, we multiply the mass of carbon by 1 (since there is one carbon atom) and the mass of oxygen by 2 (for two oxygen atoms).
• Add these values to get the compound's molar mass. For CO₂, the calculation would be 12.01 g/mol (for C) + (2 × 16.00 g/mol for O), giving us its molar mass of 44.01 g/mol.

Once you have the compound's molar mass, you can convert between mass and moles, a critical step for numerous calculations in chemistry, including using the ideal gas law.

Temperature Conversion

Temperature conversion between Celsius and Kelvin is a simple but essential step in scientific calculations. Kelvin (K) is the unit of thermodynamic temperature used in the physical sciences, and it is especially significant in equations that deal with physical laws, like the ideal gas law. Since Kelvin is an absolute temperature scale, 0 K (also known as 'absolute zero') is the point where particles theoretically stop moving.

To convert Celsius to Kelvin, which is crucial for our problem, you add 273.15 to the Celsius temperature. Here’s why this addition is necessary:

• The degree size in Kelvin is the same as in Celsius, so we don't need to adjust for scale differences.
• 0 K is absolute zero, while 0°C is the freezing point of water. The freezing point of water in Kelvin is 273.15 K.
• Therefore, the conversion is simply about shifting the scale up by 273.15 units to match the Kelvin scale with the Celsius scale at the freezing point of water.

Remember, always use Kelvin in gas law calculations, because pressure, volume, and the amount of gas are directly compared to energy, which requires an absolute temperature for accurate relationships in these scenarios.