Question

A gas has an initial volume of \(67.5 \mathrm{~mL}\) and an initial temperature of \(315 \mathrm{~K}\). What is its new volume if temperature is changed to 244 K? Assume pressure and amount are held constant.

Short answer

The new volume is approximately 52.3 mL.

Step-by-step solution

Understanding the Relationship

Since pressure and the amount of gas are constant, we will apply Charles's Law. Charles's Law states that the volume of a gas is directly proportional to its temperature at constant pressure, i.e., \( \frac{V_1}{T_1} = \frac{V_2}{T_2} \).

Identify Known Values

Here, the initial volume \( V_1 = 67.5 \text{ mL} \) and the initial temperature \( T_1 = 315 \text{ K} \). The new temperature \( T_2 = 244 \text{ K} \). We need to find \( V_2 \).

Set Up the Equation

According to Charles's Law, set up the equation as \( \frac{67.5}{315} = \frac{V_2}{244} \).

Solve for the New Volume

Rearrange the equation to solve for \( V_2 \): \( V_2 = \frac{67.5 \times 244}{315} \).

Calculate the New Volume

Perform the calculation: \( V_2 = \frac{16470}{315} \approx 52.3 \text{ mL} \).

Key concepts

Gas Laws

Gas laws describe how gases behave under different conditions, such as changes in pressure, temperature, and volume. These laws are instrumental in understanding and predicting gas behavior. One of the foundational gas laws is Charles's Law, which examines the relationship between temperature and volume of a gas when pressure is kept constant.

Gas laws are based on experimental observations and can be expressed mathematically, helping us predict how a gas will react when subjected to various conditions. They include:

• Charles's Law - Focuses on temperature and volume relationships at constant pressure.
• Boyle's Law - Describes the pressure and volume relationship at constant temperature.
• Avogadro's Law - Relates the volume and amount of gas at constant temperature and pressure.
• Ideal Gas Law - Combines all three above into a single equation: \( PV = nRT \).

By understanding these gas laws, students and scientists can accurately predict gas behavior, which is vital in many scientific and industrial processes.

Temperature and Volume Relationship

Charles's Law specifically addresses the direct relationship between the temperature and volume of a gas. If the temperature of a gas increases, its volume increases as well—provided the pressure remains constant. This is because heat adds energy to gas molecules, causing them to move faster and spread out more, thus increasing volume.

Mathematically, this relationship is given by:

• \( V \propto T \) (at constant pressure).
• The law can be rearranged to \( \frac{V_1}{T_1} = \frac{V_2}{T_2} \), where \( V \) is volume and \( T \) is temperature.

In the given exercise, you can see how this law helps compute the new volume of a gas when its temperature changes. The gas's initial volume and temperature are known, and by applying this relationship, we calculate the new volume under a different temperature.

Ideal Gas Behavior

Ideal gas behavior assumes that gas molecules are point masses with no volume, and they undergo perfectly elastic collisions. Gases behave ideally under high temperature and low pressure, where intermolecular forces are minimal, and the volume occupied by the gas molecules themselves is negligible compared to the volume the gas occupies.

However, real gases can deviate from ideal behavior, especially at low temperatures and high pressures where intermolecular forces become significant and the volume of molecules cannot be ignored. Despite these deviations, gas laws, including Charles's Law, assume ideal gas behavior to simplify calculations because:

• They provide good approximations for many gases under normal conditions.
• They help simplify complex gas behaviors into manageable equations.

By understanding the ideal gas assumptions, it becomes easier to grasp why gas laws generally work, even if they don't account for every detail present in real gases.