Question

A sample of gas at \(35^{\circ} \mathrm{C}\) and 1 atmospheric pressure occupies a volume of 3.75 1. At what temperature should the gas be kept, if it is required to reduce the volume to \(3.01\) at the same pressure? (a) \(-26.6^{\circ} \mathrm{C}\) (b) \(0^{\circ} \mathrm{C}\) (c) \(3.98^{\circ} \mathrm{C}\) (d) \(28^{\circ} \mathrm{C}\)

Short answer

(a) -26.6^{\textdegree}C

Step-by-step solution

Convert Initial Temperature to Kelvin

Convert the initial temperature from Celsius to Kelvin using the formula: Kelvin = Celsius + 273.15. For the given temperature, 35°C becomes: 35 + 273.15 = 308.15 K.

Use Charles's Law Equation

Charles's Law states that for a given mass of an ideal gas at constant pressure, the volume is directly proportional to its temperature in Kelvin. The equation is: V1/T1 = V2/T2. V1 is the initial volume, T1 is the initial temperature in Kelvin, V2 is the final volume, and T2 is the final temperature in Kelvin.

Rearrange Charles's Law to Solve for Final Temperature

The final temperature can be calculated by rearranging Charles's Law to solve for T2: T2 = (V2 * T1) / V1.

Insert Given Values

Insert the given values into the rearranged equation: T2 = (3.01 L * 308.15 K) / 3.75 L.

Calculate the New Temperature in Kelvin

After inserting the values, calculate T2 which is the final temperature in Kelvin.

Convert Final Temperature Back to Celsius

Convert the final temperature from Kelvin back to Celsius using the formula: Celsius = Kelvin - 273.15.

Choose the Correct Answer

Compare the calculated Celsius temperature with the options given in the exercise to find the correct one.

Key concepts

Ideal Gas Law

The ideal gas law is a fundamental principle that relates the pressure, volume, temperature, and the amount of gas within a closed system. Represented by the equation \( PV = nRT \), where \(P\) stands for pressure, \(V\) is the volume, \(n\) is the number of moles, \(R\) is the universal gas constant, and \(T\) is temperature in Kelvin. This law is crucial for understanding how gases behave under different conditions.

Within the context of Charles’s Law problems, the ideal gas law helps to understand that if the amount of gas (\textbackslash \textbackslash n\textbackslash \textbackslash ) and pressure (\textbackslash \textbackslash P\textbackslash \textbackslash) are constant, the relationship between volume and temperature can be isolated. Thus, when we solve Charles’s Law problems, we’re applying a component of the ideal gas law specifically focusing on volume-temperature relationship at a constant amount of gas and pressure. By understanding the ideal gas law, students can appreciate how these individual gas laws fit into the broader picture of gas behavior.

Recognizing the direct connection between \(V\) and \(T\) in Charles’s Law is a subset of this larger equation, allowing us to predict how changes in temperature can affect gas volume when the moles of gas and pressure are held constant. When solving related problems, this overarching concept enables a clear grasp of why and how gases expand with heat or contract with cold.

Gas Volume Temperature Relationships

Charles's Law describes the relationship between the volume and temperature of a gas. It states that at a constant pressure, the volume of a given amount of gas is directly proportional to its temperature measured in Kelvin. This can be written as \( \frac{V_1}{T_1} = \frac{V_2}{T_2} \), where \(V_1\) and \(V_2\) are the initial and final volumes, and \(T_1\) and \(T_2\) are the initial and final temperatures, respectively.

This means that if you increase the temperature of the gas (in Kelvin), the volume will also increase if the pressure remains unchanged. Conversely, if the temperature decreases, so will the volume. An intuitive understanding of this concept can be easily visualized: imagine a balloon expanding when heated and shrinking when cooled.

In our exercise, since the pressure and the amount of gas remain constant, we used Charles’s Law to determine the new volume of gas at a different temperature. The proportional relationship allows us to solve for the unknown temperature given the change in volume. This concept is pivotal for a variety of scientific calculations, such as predicting the behavior of gases in different weather conditions or designing engines and other devices that depend on gas expansion and contraction.

Temperature Conversion

Temperature conversion between Celsius and Kelvin is an essential process in solving gas law problems, especially when using Charles's Law. In the metric system, temperature is typically measured in degrees Celsius (°C), but the Kelvin (K) scale is used when working with gas laws because it starts at absolute zero, the theoretical point where particles have minimal kinetic energy.

The conversion formulae used are \( K = \degree C + 273.15 \) to convert from Celsius to Kelvin, and \( \degree C = K - 273.15 \) to convert from Kelvin to Celsius. It is important to convert temperatures to the Kelvin scale before using gas law equations because these laws are derived based on the direct proportionality of volume and temperature in Kelvin. Using Celsius would not accurately reflect this relationship as it is not an absolute scale.

Understanding how to convert temperatures accurately is crucial. It enables students to correctly apply gas laws to predict how gas properties change with temperature and to communicate findings in scientific contexts where Kelvin is the standard. For instance, in our exercise, we converted the given Celsius temperature to Kelvin to apply Charles's Law, and then reconverted the derived temperature back to Celsius to match the provided answer choices.