Question

An expandable vessel contains \(729 \mathrm{~mL}\) of gas at \(22 \mathrm{C}\). What volume will the gas sample in the vessel have if it is placed in a boiling water bath \((100 . \quad \mathrm{C}) ?\)

Short answer

The final volume of the gas, when placed in a boiling water bath at 100°C, is approximately \(923.83 \mathrm{~mL}\).

Step-by-step solution

Understand the gas law involved - Charles' Law

Charles' Law is given by the formula: \[V_1/T_1 = V_2/T_2\] In the formula, \(V_1\) and \(V_2\) represents the initial and final volumes, and \(T_1\) and \(T_2\) represent the initial and final temperatures in Kelvin. Remember that the temperatures must be in Kelvin for this law.

Convert given temperatures to Kelvin

First, convert the given temperatures into Kelvin: Initial temperature: 22°C = \(22 + 273.15 = 295.15 K\) Final temperature: 100 °C = \(100 + 273.15 = 373.15 K\)

Solve for the final volume

Now that we have the given temperatures in Kelvin, let's solve the equation using the given volume and temperatures: \[\frac{V_1}{T_1} = \frac{V_2}{T_2}\] Plug in the given values: \[\frac{729}{295.15} = \frac{V_2}{373.15}\] Now solve for \(V_2\): \[V_2 = \frac{729 \times 373.15}{295.15}\] \[V_2 = 923.83\] The final volume of the gas, when placed in a boiling water bath at 100°C, is approximately \(923.83 \mathrm{~mL}\).

Key concepts

gas laws

Gas laws are essential principles that help us understand and predict the behavior of gases in various conditions. Charles' Law, one of the fundamental gas laws, establishes a relationship between volume and temperature. Specifically, it states that the volume of a gas is directly proportional to its absolute temperature when the pressure is held constant.
Charles' Law is mathematically expressed as: \[ V_1/T_1 = V_2/T_2 \] Where:

• \( V_1 \) and \( V_2 \) are the initial and final volumes.
• \( T_1 \) and \( T_2 \) are the initial and final temperatures (in Kelvin).

This equation tells us that if a gas is heated, its volume will increase, provided the pressure does not change. Conversely, cooling the gas will cause its volume to decrease under constant pressure. Using Charles' Law, you can predict how much a gas will expand or contract with a change in temperature, making it a valuable tool in various scientific and engineering applications. Understanding gas laws like Charles' Law enables us to solve practical problems and make accurate assumptions about gas behaviors in everyday life.

temperature conversion

Converting temperature units is an essential step when working with gas laws, particularly Charles' Law, as calculations must utilize the Kelvin scale. The Kelvin scale is crucial because it starts at absolute zero, where all molecular motion ceases. Hence, it provides a consistent base for temperature comparisons.
To convert temperatures from Celsius to Kelvin, a straightforward formula is applied: \[ K = °C + 273.15 \] This addition brings all values onto a scale that allows for accurate computations with various gas laws. For instance, in our exercise, the initial temperature of 22°C is converted to 295.15 K, and the final temperature of 100°C is converted to 373.15 K.
This conversion is vitally important because using Celsius or Fahrenheit would lead to incorrect results when using Charles' Law or other gas laws. Always remember: when working with gas laws, Kelvin is the key scale to bring precision and accuracy to your calculations.

Kelvin scale

The Kelvin scale is integral to thermodynamic calculations in sciences, including when dealing with gas laws. It is the Standard International (SI) unit for temperature, thus preferred in scientific calculations. This scale is unique because it defines 0 Kelvin as absolute zero, the theoretical point where no molecular kinetic energy remains in a substance.
Why is the Kelvin scale used in such calculations instead of Celsius or Fahrenheit? Because it provides a true proportional scale that maintains the accurate relationships required in scientific formulas, such as Charles' Law and other related equations.
Some key points to note about the Kelvin scale are:

• 1 Kelvin and 1 degree Celsius have the same incremental change.
• 273.15 K is equivalent to 0°C, making conversion between them straightforward.
• It eliminates the need for negative temperatures in calculations, ensuring consistency and reducing errors.

By always using Kelvin, you ensure your calculations reflect the realistic behavior of gases and thermodynamic systems. It helps avoid the pitfalls that come with using relative scales like Celsius or Fahrenheit, thus maintaining the integrity of scientific investigations and analyses.