Question

At 1 atmospheric pressure and \(0^{\circ} \mathrm{C}\), certain mass of a gas measures \(0.4\) L. Keeping the pressure constant, if the temperature is increased to \(273^{\circ} \mathrm{C}\), what will be its volume? (a) \(0.8 \mathrm{~L}\) (b) \(22.4 \mathrm{~L}\) (c) \(54.6 \mathrm{~L}\) (d) \(0.4 \mathrm{~L}\)

Short answer

The volume of the gas when increased to 273 degrees Celsius at constant pressure will be 0.8 L.

Step-by-step solution

Understand Charles's Law

Charles's Law states that for a given mass of gas at constant pressure, the volume is directly proportional to its Kelvin temperature. It can be expressed as V1/T1 = V2/T2, where V1 and T1 are the volume and temperature (in Kelvin) of the gas initially, and V2 and T2 are the volume and temperature of the gas after a change.

Convert Celsius to Kelvin

The temperature needs to be converted from Celsius to Kelvin. Since 0 degrees Celsius is equal to 273.15 Kelvin, and 273 degrees Celsius is 273 + 273.15 = 546.15 Kelvin.

Set up the equation using Charles's Law

Using Charles's Law, set up the ratio V1/T1 = V2/T2. Plugging the values into the equation gives us 0.4/273.15 = V2/546.15.

Solve for V2

Cross-multiply and divide to solve for V2: V2 = (0.4 * 546.15) / 273.15. Calculate V2 to get the new volume.

Calculate the new volume

V2 = (0.4 * 546.15) / 273.15 = 0.8 L. Therefore, the volume of the gas at 273 degrees Celsius and 1 atmospheric pressure is 0.8 L.

Key concepts

Gas Laws

The behavior of gases is governed by several fundamental principles known as gas laws. These laws describe the quantitative relationship between pressure, volume, temperature, and amount (moles) of a gas. Among these, Charles's Law is especially important, highlighting the volume-temperature relationship at constant pressure.

Understanding gas laws is crucial for students studying chemistry because these principles explain how gases will react under different conditions. Charles's Law, for instance, is particularly useful when predicting how a gas's volume will change with temperature. It signifies that if you heat a gas, as long as the pressure doesn't change, its volume will increase; conversely, cooling it down will cause the volume to reduce. These predictions are vital in several practical scenarios such as balloon flight, engine design, and even understanding weather patterns.

Charles's Law and other gas laws are not just theoretical—they form the basis for real-life applications in various scientific and engineering fields. With a strong grasp of these principles, students can solve problems related to gas behavior under different physical conditions.

Volume Temperature Relationship

The volume-temperature relationship, as detailed in Charles's Law, states that a gas's volume is directly proportional to its absolute temperature, provided that the pressure remains constant. This relationship is expressed mathematically as \( \frac{V_1}{T_1} = \frac{V_2}{T_2} \), where \( V_1 \) and \( T_1 \) refer to the original volume and temperature, and \( V_2 \) and \( T_2 \) to the changed volume and temperature.

When working with this relationship, it's important to remember that the temperature must always be in Kelvin units. The Kelvin scale is an absolute temperature scale, meaning it starts at absolute zero and only has positive values. This makes Kelvin the ideal unit for scientific work involving temperature. In Charles's Law, using Kelvin units ensures that proportional changes in volume and temperature will yield correct and meaningful results.

To practically apply this relationship, it is often necessary to make unit conversions. If the given temperature is in Celsius, for example, students need to convert it to Kelvin by adding 273.15. This precise conversion allows for correct application of the volume-temperature relationship and, by extension, accurate predictions and solutions for various problems related to gas behavior.

Kelvin Temperature Scale

The Kelvin temperature scale is the cornerstone of temperature measurement in the scientific community, and it plays an indispensable role in quantitative gas laws. The Kelvin scale is based on the concept of absolute zero, the theoretical temperature at which particles have minimal kinetic energy and cease to move.

The Kelvin scale is relationally identical to the Celsius scale, with each unit increment corresponding to the same temperature change. The key difference is the starting point: 0 degrees Celsius corresponds to 273.15 Kelvin. This offset ensures that there are no negative values on the Kelvin scale, which is important for calculations in thermodynamics and other areas of physical science.

When solving chemistry problems related to gas laws like Charles's Law, temperatures must be converted to Kelvin to avoid negative numbers and to provide an accurate portrayal of the energy states of particles in a gas. As a content creator, it's important to emphasize to the students the necessity of using the Kelvin scale in their calculations to ensure proper understanding and to facilitate error-free problem solving.