Question

If \(12.0 \mathrm{g}\) of \(\mathrm{O}_{2}\) is required to inflate a balloon to a certain size at \(27^{\circ} \mathrm{C},\) what mass of \(\mathrm{O}_{2}\) is required to inflate it to the same size (and pressure) at \(5.0^{\circ} \mathrm{C} ?\)

Short answer

11.12 grams of \(\mathrm{O}_2\) is required at 5.0°C.

Step-by-step solution

Identify Known Values

We have the initial mass of \(\mathrm{O}_2\) as \(12.0\, \mathrm{g}\), and the initial temperature \(T_1\) is \(27^{\circ} \mathrm{C}\). The final temperature \(T_2\) is \(5.0^{\circ} \mathrm{C}\). These temperatures need to be converted to Kelvin for calculations: \(T_1 = 27 + 273 = 300\,\mathrm{K}\) and \(T_2 = 5 + 273 = 278\,\mathrm{K}\).

Use the Ideal Gas Law

Assume the balloon fills at constant volume and pressure. Use the relationship \(\frac{n_1}{T_1} = \frac{n_2}{T_2}\), where \(n\) is the number of moles of \(\mathrm{O}_2\). Since \(n = \frac{\text{mass}}{\text{molar mass}}\), we substitute to find the mass relation: \(\frac{\text{mass}_1}{T_1} = \frac{\text{mass}_2}{T_2}\).

Calculate the Molar Mass of \(\mathrm{O}_2\)

The molar mass of \(\mathrm{O}_2\) is \(32.0\, \mathrm{g/mol}\). This will be needed to relate the mass of \(\mathrm{O}_2\) to moles.

Solve for Mass at New Temperature

From the equation \(\frac{12.0}{300} = \frac{\text{mass}_2}{278}\), solve for \(\text{mass}_2\): \(\text{mass}_2 = \frac{12.0 \times 278}{300} = 11.12 \, \mathrm{g}\).

Verify the Relationship

Check the mass relation \(\text{mass}_2 < \text{mass}_1\) since the temperature has decreased, which means fewer molecules are needed to maintain volume at lower temperature.

Key concepts

Gas Laws

The gas laws are fundamental principles that describe how gases behave under different conditions of temperature, volume, and pressure. These laws are essential for predicting how a gas will change when any one of these variables is altered. One of the most important gas laws is the Ideal Gas Law, which is usually expressed as \( PV = nRT \), where \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the ideal gas constant, and \( T \) is temperature in Kelvin.

When dealing with situations where the volume and pressure of a gas are constant, we can simplify our calculations using a relationship derived from the Ideal Gas Law: \( \frac{n_1}{T_1} = \frac{n_2}{T_2} \). This relationship connects the number of moles of the gas to its absolute temperature. It helps predict how much gas is needed when temperatures change, ensuring the volume remains the same.

Temperature Conversion

In physics and chemistry, it’s often necessary to convert temperatures between Celsius and Kelvin. This conversion is critical because many scientific equations, including those involving gases, require temperature in Kelvin. The Kelvin scale is an absolute temperature scale starting at absolute zero, where there is no kinetic energy in a substance's molecules.

To convert from Celsius to Kelvin, simply add 273.15 to the Celsius temperature: \( T_{\text{Kelvin}} = T_{\text{Celsius}} + 273.15 \). For our problem: the initial temperature \( 27^{\circ} \mathrm{C} \) becomes \( 300 \mathrm{K} \), and \( 5^{\circ} \mathrm{C} \) converts to \( 278 \mathrm{K} \). This conversion is precise and avoids any errors that might occur when working with different temperature scales.

Molar Mass Calculation

Molar mass is an essential concept for relating the mass of a substance to the amount of moles, a counting unit used to measure chemical substances. The molar mass of a substance is expressed in grams per mole (\( \mathrm{g/mol} \)), and it tells us the mass of one mole of that substance. This is particularly useful when applying the Ideal Gas Law, as it allows for the conversion between mass and moles.

For \( \mathrm{O}_2 \), the molar mass is calculated by adding the atomic masses of two oxygen atoms. Each oxygen atom has an atomic mass of approximately \( 16.0 \), so the molar mass of \( \mathrm{O}_2 \) is \( 32.0 \, \mathrm{g/mol} \). This calculation is important in order to use the Ideal Gas Law effectively and determine how the mass changes with temperature.

Mole Concept

The mole concept is a fundamental idea in chemistry, allowing scientists to relate the measurable quantities of substances in laboratory settings. A mole is defined as \( 6.022 \times 10^{23} \) entities (atoms, molecules, or other particles) and provides a bridge between atoms and the macroscopic amounts of material we handle in the laboratory.

The relationship between moles and mass is given by \( n = \frac{\text{mass}}{\text{molar mass}} \), where \( n \) is the number of moles. This formula can be used to convert between the mass of a substance and the amount in moles, which was crucial in solving our problem. By understanding how moles relate to temperature and mass, we can predict the behavior of gases under different conditions using the Ideal Gas Law.