Question

A certain gas occupies a volume of \(100 \mathrm{ml}\) at a temperature of \(20^{\circ} \mathrm{C}\). What will its volume be at \(10^{\circ} \mathrm{C}\), if the pressure remains constant?

Short answer

The volume of the gas at a temperature of \(10^{\circ} \mathrm{C}\) (283.15 K) and constant pressure is approximately 96.6 ml.

Step-by-step solution

Write the given information and convert temperature to Kelvin

Given data Initial volume of gas, V1 = 100 ml Initial temperature, T1 = 20°C Final temperature, T2 = 10°C First, convert the given temperatures from Celsius to Kelvin. To do this, add 273.15 to each Celsius temperature value. T1 (in K) = 20°C + 273.15 = 293.15 K T2 (in K) = 10°C + 273.15 = 283.15 K

Apply Charles' law formula

Charles' law states that for a given amount of gas at constant pressure, the volume is directly proportional to the temperature in Kelvin. Mathematically, the law can be written as: \( \frac{V1}{T1} = \frac{V2}{T2} \) Where V1 and T1 are the initial volume and temperature, and V2 and T2 are the final volume and temperature.

Solve for the final volume (V2)

Use the formula from step 2 and plug in the values for V1, T1, and T2, then solve for V2. \( \frac{100 \, \mathrm{ml}}{293.15 \, \mathrm{K}} = \frac{V2}{283.15 \, \mathrm{K}} \) To find V2, multiply both sides of the equation by T2: \( V2 = \frac{100\, \mathrm{ml} \times 283.15 \, \mathrm{K}}{293.15 \, \mathrm{K}} \) Now, calculate the final volume: V2 ≈ \( \frac{100\, \mathrm{ml} \times 283.15 \, \mathrm{K}}{293.15 \, \mathrm{K}} \) = 96.6 ml

Report the final volume

The volume of the gas at a temperature of 10°C (283.15 K) and constant pressure is approximately 96.6 ml.

Key concepts

Gas Volume Calculation

Understanding gas volume calculation is essential when studying gases and their behavior under different conditions. In the scenario provided, we have a fixed amount of gas and we want to determine how its volume changes with temperature, while keeping the pressure constant. This is possible thanks to Charles' Law, which we'll explore later.

In our exercise, the initial volume of the gas is given as 100 ml. We need to find out the new volume at a different temperature. By using the formula from Charles' Law, we can relate the initial and final temperatures and volumes. It’s important to note that we first need to ensure all temperatures are in Kelvin for this calculation to be accurate.
When calculating the volume changes, ensure to use correct mathematical operations, keeping track of the units during the calculations.

Temperature Conversion to Kelvin

Temperature conversion to Kelvin is a crucial step when dealing with gas laws like Charles' Law because temperature must be in an absolute scale. The Kelvin scale is used here because it starts at absolute zero, where molecular motion stops.

To convert a temperature from Celsius to Kelvin, you simply add 273.15 to the Celsius value. For instance, in our example, the initial temperature of 20°C becomes 293.15 K after conversion. Similarly, the final temperature of 10°C becomes 283.15 K.

Remembering this conversion is essential, especially when working with any gas law, as dealing with Celsius could result in incorrect results. All temperature-related calculations in physics and chemistry should work with Kelvin to maintain consistency and accuracy.

Ideal Gas Law

Although the problem specifically applies Charles' Law, it's helpful to understand where it fits within the broader framework of the ideal gas law. The ideal gas law is expressed as:\[ PV = nRT \]where:- \( P \) is the pressure of the gas,- \( V \) is the volume,- \( n \) is the number of moles,- \( R \) is the ideal gas constant,- \( T \) is the temperature in Kelvin.
Charles' Law is a derivative of the ideal gas law, assuming pressure and the quantity of the gas (n) are constant. This simplifies to:\[ \frac{V}{T} = \text{constant} \]This relationship shows us that as the temperature of a gas increases, so does its volume, provided that its pressure does not change. The ideal gas law thus provides a complete picture of how gases behave under various conditions, with Charles' Law being a specific application when focusing on volume and temperature.