Problem 100 - Introductory Chemistry Atoms First

Question

A chemistry student realizes she has forgotten the value of \(R\) and needs to determine it experimentally. She decides to measure the mass, volume, pressure, and temperature of a sample of carbon dioxide gas. She finds that \(4.505 \mathrm{~g}\) of the gas occupies \(2.50 \mathrm{~L}\) at \(23^{\circ} \mathrm{C}\) and \(0.9960 \mathrm{~atm}\). Calculate the value of \(R\) she determines from these data.

Step-by-Step Solution

Answer

The experimentally determined gas constant R value can be calculated using the ideal gas law equation and the given data. Convert mass to moles, temperature to Kelvin, and use the given pressure and volume in atm and L respectively. Plug these values into the ideal gas law equation (PV = nRT) and solve for R. The resulting R value is approximately 0.0843 L atm / (mol K).

Step 1: Convert mass to moles

First, we need to convert the mass of the carbon dioxide gas into moles. To do this, we will use the molar mass of carbon dioxide, which is 44.01 g/mol (12.01 g/mol for carbon and 2 × 16.00 g/mol for oxygen). The moles of carbon dioxide can be calculated using the following formula: n = mass / molar mass where n is the number of moles, mass is given as 4.505 g, and molar mass is 44.01 g/mol. n = 4.505 g / 44.01 g/mol

Step 2: Calculate moles

By calculating the moles, we get: n = 4.505 g / 44.01 g/mol = 0.1023 mol

Step 3: Convert temperature to Kelvin

Next, we need to convert the temperature given in Celsius, to Kelvin. We can do this by adding 273.15 to the given temperature: T(K) = T(°C) + 273.15 T(K) = 23°C + 273.15 = 296.15 K

Step 4: Convert pressure to atm

The pressure is given in atm, so we don't need to convert it. P = 0.9960 atm

Step 5: Convert volume to L

The volume is given in L, so we don't need to convert it. V = 2.50 L

Step 6: Solve for R using the Ideal Gas Law

Now that we have all our values in the correct units, we can plug them into the ideal gas law equation (PV = nRT) and solve for R: 0.9960 atm × 2.50 L = 0.1023 mol × R × 296.15 K R = (0.9960 atm × 2.50 L) / (0.1023 mol × 296.15 K)

Step 7: Calculate R

By calculating R, we get: R = (0.9960 atm × 2.50 L) / (0.1023 mol × 296.15 K) = 0.0843 L atm / (mol K) The experimentally determined R value is 0.0843 L atm / (mol K).

Key Concepts

Universal Gas Constant

The Universal Gas Constant, often symbolized as \( R \), is a crucial factor in the Ideal Gas Law equation, \( PV = nRT \). This constant relates the energy scale in chemistry with the temperature scale and participates in equations that involve gases. It helps connect different measurable properties of gases like pressure, volume, temperature, and moles. When solving gas problems, you may encounter different values of \( R \) depending on the units used for pressure and volume:

• When using atm (atmospheres) for pressure and L (liters) for volume, \( R \) is approximately 0.0821 L atm / (mol K).
• For pressure in pascals (Pa) and volume in cubic meters (m³), \( R \) is approximately 8.314 J / (mol K).

In our solution, since pressure is in atm and volume is in liters, we determined \( R \) experimentally to be 0.0843 L atm / (mol K).

Molar Mass

Molar Mass is the weight of one mole of a substance, usually expressed in grams per mole (g/mol). It's like a bridge that connects the mass of a sample to the number of moles in that sample, which is essential for many calculations in chemistry. To determine the moles from a given mass, use the formula: \[ n = \frac{\text{mass}}{\text{molar mass}} \]For carbon dioxide (\( CO_2 \)), the molar mass is calculated based on its constituent atoms:- Carbon (C): 12.01 g/mol- Oxygen (O): 16.00 g/mol (since there are two oxygens, this amounts to: 2 * 16.00 g/mol)Thus, the molar mass of \( CO_2 \) is \( 12.01 + (2 \times 16.00) = 44.01 \) g/mol.In the exercise, converting 4.505 grams of \( CO_2 \) to moles using the molar mass finds us approximately 0.1023 moles.

Gas Laws

Gas laws are a series of scientific principles that describe the behavior of gases. These include well-known laws like Boyle's Law, Charles's Law, and Avogadro's Law, all of which contribute to the Ideal Gas Law.The Ideal Gas Law is expressed as \( PV = nRT \), combining the variables:- \( P \): Pressure in atm or another unit- \( V \): Volume in liters or cubic meters- \( n \): Number of moles- \( R \): Universal Gas Constant- \( T \): Temperature in KelvinThese parameters allow the description of the state of an ideal gas and predict how it will change under varying conditions.In the exercise, this law was used to understand how the student could determine \( R \) experimentally by measuring physical properties and calculating the number of moles from the known molar mass.

Temperature Conversion

Temperature conversion is essential when working with gas laws because temperature needs to be in Kelvin for the equations to make sense. The Kelvin scale is an absolute temperature scale used in scientific equations because it avoids negative numbers and starts at absolute zero, where matter has minimal energy.To convert temperature from Celsius to Kelvin, utilize this simple formula:\[ T(K) = T(°C) + 273.15 \]This ensures that all temperatures used in calculations are positive and conform to the requirements of the Ideal Gas Law.In the given exercise, the temperature of 23°C was converted to Kelvin, resulting in approximately 296.15 K, facilitating the proper use of the gas law equations.