Question

A gas occupies a volume of \(1.0\) liter at a temperature of \(27^{\circ} \mathrm{C}\) and 500 Torr pressure. Calculate the volume of the gas if the temperature is changed to \(60^{\circ} \mathrm{C}\) and the pressure to 700 Torr.

Short answer

The final volume of the gas when the pressure is changed to 700 Torr and the temperature to \(60^{\circ} \mathrm{C}\) is approximately \(0.792\, L\).

Step-by-step solution

Convert temperatures to Kelvin

Since the temperatures in the given exercise are in Celsius, we need to convert them to Kelvin before using the combined gas law formula. To convert Celsius to Kelvin, simply add 273.15 to the Celsius temperature. \(T_1 = 27^{\circ}\mathrm{C} + 273.15 = 300.15 K\) \(T_2 = 60^{\circ} \mathrm{C} + 273.15 = 333.15 K\)

Apply the combined gas law formula

Now that we have the temperatures in Kelvin, we can use the combined gas law formula: \(\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}\) We know the initial and final pressures, temperatures, and initial volume and want to find the final volume \(V_2\). So we can rearrange the formula to solve for \(V_2\): \(V_2 = \frac{P_1V_1T_2}{T_1P_2}\)

Substitute the values and calculate the final volume

Now we can substitute the given values into the formula and calculate the final volume: \(V_2 = \frac{(500\,\text{Torr}) (1.0\, \text{L}) (333.15\, K)}{(300.15\, K) (700\, \text{Torr})}\) \(V_2 = \frac{166575\, \text{Torr} \cdot \text{L} \cdot K}{210105\, \text{Torr} \cdot K}\) \(V_2 = 0.792\, L\) So the final volume of the gas when the pressure is changed to 700 Torr and the temperature to \(60^{\circ} \mathrm{C}\) is approximately \(0.792\, L\).

Key concepts

Understanding Gas Laws

Gas laws are a fundamental aspect of physical science, providing insights into the behavior of gases under various conditions. These laws relate the volume, pressure, temperature, and quantity (moles) of a gas to one another.

One crucial gas law is the combined gas law, which is a combination of three simpler laws: Boyle’s Law, Charles’s Law, and Gay-Lussac’s Law. Boyle’s Law states that for a fixed amount of gas at constant temperature, the volume of a gas is inversely proportional to its pressure. Charles’s Law deduces that at a constant pressure, the volume of a gas is directly proportional to its temperature (measured in Kelvin). Lastly, Gay-Lussac's Law asserts that the pressure of a gas is directly proportional to its temperature at a constant volume.

The combined gas law formula is particularly useful when dealing with scenarios where both temperature and pressure change. It allows you to predict the result of these changes on the volume of a gas without keeping mass or the amount constant.

• It's crucial to remember all gas law calculations require absolute temperatures, hence the need for temperature conversion.
• Also, accurate measurement units need to be maintained for pressure and volume.

Understanding how these properties interact is key to mastering gas laws and solving related problems effectively.

Temperature Conversion Basics

Temperature conversion between Celsius (°C) and Kelvin (K) is a critical step when using gas laws, as calculations must be performed using the absolute temperature scale — Kelvin.

To convert from Celsius to Kelvin, you add 273.15 to the Celsius temperature, thus establishing the necessary base for accurately using the combined gas law formula. The equation is simple:
K = °C + 273.15.

Why Kelvin, Not Celsius?

The Kelvin scale starts at absolute zero, the point at which particles (theoretically) cease motion, making it the most suitable and consistent measure for scientific calculations involving temperature. In contrast, the Celsius scale is based on the properties of water, with 0°C as the freezing point and 100°C as the boiling point under standard atmospheric pressure. For gases, their behaviors at these points are not linear, which is why the Kelvin scale is preferred.

The Pressure-Volume Relationship

The pressure-volume relationship is an important aspect when working with gases, described by Boyle’s Law as part of the combined gas law. It helps us understand how gases will compress or expand with changes in pressure, assuming the temperature and amount of gas remain constant.

When the pressure increases, the volume decreases and vice versa, which is referred to as an inverse relationship. To see this in action, consider pressing down on a syringe—the air inside compresses as you apply pressure, reducing the volume. In mathematical terms, the product of the initial pressure and volume will always be equal to the product of the final pressure and volume, assuming no changes in temperature or amount of gas, which is depicted as:
P1 * V1 = P2 * V2.

In solving problems involving the pressure-volume relationship, it's also essential to maintain consistent units for pressure, whether it's atmospheres (atm), Pascals (Pa), Torr, or millimeters of mercury (mmHg), to ensure accuracy in computations and clarity in understanding the behavior of gases.