Question

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

Short answer

The gas sample in the vessel will have a volume of approximately \(920 \mathrm{mL}\) when it is placed in a boiling water bath at \(100^\circ\text{C}\).

Step-by-step solution

Convert temperatures to Kelvin

Convert the initial and final temperatures from Celsius to Kelvin. To do this, add 273.15 to both temperatures. Initial temperature (T_i): \(T_i = 22^\circ\text{C} + 273.15 = 295.15\text{K} \) Final temperature (T_f): \(T_f = 100^\circ\text{C} + 273.15 = 373.15\text{K} \)

Write down the Charles' Law formula

Charles' Law states that the volume of an ideal gas is directly proportional to its temperature, given that the pressure and amount of gas are constant. The formula is: \( \frac{V_i}{T_i} = \frac{V_f}{T_f} \) Where \(V_i\) is the initial volume, \(T_i\) is the initial temperature, \(V_f\) is the final volume, and \(T_f\) is the final temperature. All temperatures are in Kelvin.

Input the known values into the formula

The initial volume (\(V_i\)) and initial temperature (\(T_i\)) are given in the problem, and we have calculated the final temperature (\(T_f\)). Plug these values into the Charles' Law formula: \( \frac{729 \mathrm{mL}}{295.15\text{K}} = \frac{V_f}{373.15\text{K}} \)

Solve for the final volume \((V_f)\)

Solve for the final volume (\(V_f\)), by cross multiplying and dividing: \( V_f = \frac{729 \mathrm{mL} \times 373.15\text{K}}{295.15\text{K}} \) Now, calculate the final volume: \( V_f = \frac{272252.35 \mathrm{mL\cdot K}}{295.15\text{K}} \) \( V_f ≈ 922.42 \mathrm{mL} \)

Round the result to the appropriate number of significant figures

Since the problem contains only two significant figures in both initial values (729 mL and \(22^\circ\text{C}\)), the final volume should also have two significant figures. The final volume, rounded to two significant figures, is: \(V_f ≈ 920 \mathrm{mL}\) So, the gas sample in the vessel will have a volume of approximately 920 mL when it is placed in a boiling water bath at \(100^\circ\text{C}\).

Key concepts

Ideal Gas Law

The ideal gas law is a fundamental equation in chemistry and physics that describes the behavior of an ideal gas. It merges several individual gas laws, including Charles' Law, into a single expression. The formula for the ideal gas law is given by:
\( PV = nRT \)
where

• \(P\) stands for the pressure of the gas,
• \(V\) is the volume,
• \(n\) is the number of moles,
• \(R\) is the ideal gas constant, and
• \(T\) is the temperature in Kelvin.

Charles' Law is one aspect of the ideal gas law, more specifically related to volume and temperature at constant pressure and amount of gas. The ideal gas law simplifies the study of gases by combining several relationships into one, allowing us to predict how a gas will change under various conditions. When using the ideal gas law, it is essential to be mindful of the units used for pressure, volume, and temperature, and to use absolute temperature in Kelvin.

Temperature Conversion

Temperature conversion is vital in gas law problems since most gas law equations require temperature to be in Kelvin rather than Celsius or Fahrenheit. To convert a temperature from Celsius to Kelvin, you add 273.15 to the Celsius temperature. This is because

• the Kelvin scale starts at absolute zero, unlike the Celsius scale which is based on the freezing and boiling points of water,
• each unit on the Kelvin scale is the same size as the Celsius scale, making the conversion straightforward.

To illustrate, taking our example, the conversion process for an initial temperature of \(22^\text{C}\) would be: \( T_\text{i} = 22^\text{C} + 273.15 = 295.15K \). Accurate temperature conversion is crucial, as incorrect temperatures would result in erroneous calculations in any gas-related problems.

Kelvin Scale

The Kelvin scale is an absolute temperature scale used in the scientific measurement of temperature. It is crucial in thermodynamics and the study of gases as it starts at absolute zero, which is the theoretical point at which all molecular motion ceases. Thus, the Kelvin scale allows scientists and engineers to measure temperature in an absolute sense, eliminating the negative numbers associated with the Celsius and Fahrenheit scales.

The significance of using the Kelvin scale for gases is based on the direct proportionality between temperature and the kinetic energy of gas particles. As temperature increases, so does the kinetic energy and, therefore, the movement of particles.

It is crucial to use the Kelvin scale in calculations involving the ideal gas law or its components such as Charles' Law, as it ensures consistent and accurate results.

Significant Figures

Significant figures play a crucial role in the accuracy and precision of calculations, especially in scientific contexts. They indicate the number of meaningful digits in a number, starting with the first non-zero digit. Here’s why they matter:

• Significant figures reflect the precision of the measurement tools used to obtain data,
• they help determine the level of uncertainty in calculations,
• in computations, the result should not be more precise than the least precise measurement.

Applying this principle, we pay attention to the significant figures in input values during calculations. As in the given exercise, our values (729 mL and \(22^\text{C}\)) have two significant figures, hence our final volume result should also be reported with two significant figures: \( V_f \approx 920 \text{mL}\). By rounding our answer accordingly, we maintain the integrity of the data's precision throughout our calculations.