Problem 66 - Chemistry An Atoms First Approach

Question

A student adds \(4.00 \mathrm{g}\) of dry ice (solid \(\mathrm{CO}_{2}\) ) to an empty balloon. What will be the volume of the balloon at STP after all the dry ice sublimes (converts to gaseous \(\mathrm{CO}_{2}\) )?

Step-by-Step Solution

Answer

The volume of the balloon after all the dry ice sublimes will be approximately \(2.1 \mathrm{L}\) at STP.

Step 1: Calculate moles of CO₂

To calculate the moles of CO₂, we need to use the given mass and the molar mass of CO₂. The molar mass of CO₂ is 44.01 g/mol. Moles of CO₂ = Mass of CO₂ / Molar mass of CO₂ n = 4.00 g / 44.01 g/mol n ≈ 0.0909 mol

Step 2: Use Ideal Gas Law to find Volume

Now we can use the ideal gas law to find the volume, V. At STP, the temperature, T, is 0°C (or 273.15 K) and the pressure, P, is 1 atm. We also know the gas constant, R: 0.0821 L*atm/(mol*K). The equation is written as: PV = nRT First, we should isolate the volume (V) on one side of the equation: V = nRT / P Now, we can plug in the values we know: V = (0.0909 mol) (0.0821 L*atm/(mol*K)) (273.15 K) / (1 atm) V ≈ 2.05 L

Step 3: Round the answer

Finally, we should round the answer to an appropriate number of significant figures. In this case, 2 significant figures are appropriate, since the mass of dry ice given in the problem has 2 significant figures: V ≈ 2.1 L #Answer# So, the volume of the balloon after all the dry ice sublimes will be approximately 2.1 liters at STP.

Key Concepts

STP Conditions

Standard Temperature and Pressure, abbreviated as STP, is a set of conditions used in chemistry to allow scientists to compare experimental data with theoretical calculations and other experiments. These conditions simplify calculations and ensure results are consistent across different studies.

At STP, the temperature is defined as 0°C, which is equivalent to 273.15 Kelvin. The pressure is set at 1 atmosphere (atm). These standardized conditions are crucial when using gas laws, such as the Ideal Gas Law.

• The temperature, 0°C or 273.15 K, allows chemists to have a common reference point.
• The pressure of 1 atm represents an average atmospheric pressure at sea level.

When you're working with gases, knowing the conditions is essential because the volume that a gas occupies can change significantly with different temperatures and pressures. Under STP, the volume of one mole of any ideal gas is always about 22.4 liters, which helps greatly when calculating gas volumes in reactions.

Molar Mass Calculations

The molar mass of a substance is the mass of one mole of its molecules. It's a fundamental property used in various chemical calculations. To find the molar mass, you need to sum the atomic masses of all the atoms in a molecular formula.

For example, with carbon dioxide (CO₂):

• Carbon (C) has an atomic mass of about 12.01 g/mol.
• Oxygen (O) has an atomic mass of about 16.00 g/mol.
• CO₂ has one carbon atom and two oxygen atoms.
• So, the molar mass is 12.01 + 2(16.00) = 44.01 g/mol.

Once you have the molar mass, you can quickly convert between mass and moles using the formula: \[ \text{moles} = \frac{\text{mass in grams}}{\text{molar mass in g/mol}} \]This conversion is pivotal in using the Ideal Gas Law to find volumes at STP conditions or in any other chemical calculation requiring the mole concept.

Significant Figures

Significant figures are essential in scientific calculations because they reflect the precision of measurements. They help indicate the certainty of measured values in experiments and calculations.

In general, the number of significant figures in a number includes all the known digits plus one estimated digit.

• For a measurement given as 4.00 g, there are three significant figures because all zeros following the decimal and non-zero number are considered significant.
• When performing calculations, your final answer should reflect the smallest number of significant figures in any of the values used in the calculation.

For example, if you have a value calculated with two significant figures, your result should also be rounded to two significant figures to maintain consistency and accuracy. In the given problem, although the mole calculation resulted in a more precise number, the mass of the CO₂ was given in two significant figures, so the final result was appropriately rounded to two significant figures, leading to 2.1 liters for the volume of the gas at STP. This careful rounding is vital for maintaining the integrity of the data in scientific work.