Question

A \(275-\mathrm{mL}\) sample of CO gas is collected over water at \(31^{\circ} \mathrm{C}\) and \(755 \mathrm{mmHg}\). If the temperature of the gas collection apparatus rises to \(39^{\circ} \mathrm{C}\), what is the new volume of the sample? Assume that the barometric pressure does not change.

Short answer

The new volume of the gas sample is approximately 282.23 mL.

Step-by-step solution

Understand the problem

We have a sample of gas with an initial volume of 275 mL collected over water at a temperature of \(31^{\circ} \mathrm{C}\) and a pressure of 755 mmHg. We need to find the new volume when the temperature rises to \(39^{\circ} \mathrm{C}\).

Convert temperatures to Kelvin

Since the gas laws require temperature in Kelvin, we need to convert from Celsius. Use the formula: \( T(K) = T(\degree C) + 273.15 \).For \(31^{\circ} \mathrm{C}\):\[T_1 = 31 + 273.15 = 304.15\, \mathrm{K}\]For \(39^{\circ} \mathrm{C}\):\[T_2 = 39 + 273.15 = 312.15\, \mathrm{K}\]

Apply Charles's Law

Charles's Law states \( V_1 / T_1 = V_2 / T_2 \), where \(V\) is volume and \(T\) is temperature in Kelvin. We need to solve for \(V_2\) (the new volume).

Rearrange Charles's Law to solve for \(V_2\)

Rearrange the equation from Charles's Law to find the new volume:\[V_2 = V_1 \times \frac{T_2}{T_1}\]

Calculate \(V_2\)

Substitute the known values into the equation:\[V_2 = 275 \times \frac{312.15}{304.15}\]Calculate:\[V_2 = 275 \times 1.0263 \]\[V_2 \approx 282.23 \text{ mL}\]

Determine the New Volume

The new volume of the gas when the temperature rises to \(39^{\circ} \mathrm{C}\) is approximately 282.23 mL.

Key concepts

Gas Laws

Gas laws are fundamental principles that describe how gases behave under different conditions of temperature, pressure, and volume. These laws allow us to predict changes in the state of a gas in response to changes in one of these conditions.
One of the most important gas laws used in chemical calculations is Charles's Law. This law states that if the pressure on a gas is held constant, the volume of the gas is directly proportional to its temperature in Kelvin. This means that when the temperature increases, the volume increases as well, and vice-versa, as long as the pressure doesn't change.

• Charles's Law Formula: \( V_1 / T_1 = V_2 / T_2 \)
• \(V\) represents the volume of the gas, and \(T\) represents its temperature in Kelvin.

Understanding gas laws like Charles's Law is important for predicting how gases will react to changes in environmental conditions and is widely applied in various scientific fields.

Temperature Conversion

In solving problems involving gas laws, it is crucial to convert temperatures to Kelvin, which is the absolute temperature scale used in scientific calculations. Kelvin is preferred in gas law equations because it starts at absolute zero, the point at which all molecular motion ceases.

To convert Celsius temperatures to Kelvin, you simply add 273.15:

• Formula: \( T(K) = T(\degree C) + 273.15 \)

For example:

• A temperature of \(31^{\circ} C\) converts to \( 31 + 273.15 = 304.15\, K \)
• A temperature of \(39^{\circ} C\) converts to \( 39 + 273.15 = 312.15\, K \)

Being comfortable with temperature conversion is crucial for correctly applying gas laws and ensuring accurate calculations.

Volume Calculations

Volume calculations in gas law problems require careful consideration of the relationships between temperature, pressure, and volume. When using Charles's Law to find a new volume, rearranging the equation is often necessary to solve for the unknown variable.

Here's how you handle volume calculations using Charles's Law:

• Identify initial and final temperatures and volumes.
• Ensure temperatures are in Kelvin for precision.
• Use the rearranged formula: \( V_2 = V_1 \times \frac{T_2}{T_1} \)

For instance, if the initial volume is 275 mL and the temperature changes from 304.15 K to 312.15 K:

• Calculate: \( V_2 = 275 \times \frac{312.15}{304.15} \)
• This yields \( V_2 \approx 282.23 \text{ mL} \)

Understanding each step in detail helps you accurately determine the volume changes under different thermal conditions.

Ideal Gas Behavior

The concept of ideal gas behavior is based on the Ideal Gas Law, which assumes that gases consist of small particles in random motion and that these particles interact only through elastic collisions. While real gases do exhibit some deviations, many gases behave ideally under normal conditions of temperature and pressure.

The assumptions of ideal gas behavior simplify calculations and are key to using gas laws like Charles's Law:

• Gases are composed of tiny, rapidly moving particles.
• The volume of gas particles is negligible compared to the space between them.
• No forces of attraction or repulsion between particles.
• Collisions are perfectly elastic, meaning no energy is lost.

These assumptions allow us to predict how changes in temperature might alter the volume of the gas. Understanding ideal gas behavior is essential for applying gas laws to real-world scenarios, and it often serves as a useful approximation for conducting experiments and solving problems.