Question

For each of the following sets of volume/temperature data, calculate the missing quantity. Assume that the pressure and the mass of gas remain constant. a. \(V=25.0 \mathrm{~L}\) at \(0 \quad \mathrm{C} ; V=50.0 \mathrm{~L}\) at \(? \mathrm{C}\) b. \(V=247 \mathrm{~mL}\) at 25 ' \(\mathrm{C} ; V=255 \mathrm{~mL}\) at \(? \mathrm{C}\) c. \(V=1.00 \mathrm{~mL}\) at \(2272 \mathrm{C} ; V=?\) at \(25 \mathrm{C}\)

Short answer

The missing quantities for the given sets of volume/temperature data are: a. \(546.30 K\) b. \(308.88 K\) c. \(0.117 mL\)

Step-by-step solution

Convert Celsius to Kelvin

For the given temperature, we can convert it to Kelvin using the formula: \[T(K) = T(°C) + 273.15\] For \(0 °C\), the temperature in Kelvin is: \[0 + 273.15 = 273.15 K\]

Use Charles's Law to find missing temperature

Now, we can apply Charles's Law to find the missing temperature: \[\frac{25.0}{273.15} = \frac{50.0}{T_2}\] To find \(T_2\), we can cross-multiply and divide: \[T_2 = \frac{50.0 \times 273.15}{25.0} \] \[T_2 = 546.30 K\] The missing temperature is \(546.30 K\) in Kelvin. b. Find the missing temperature for \(V = 247 mL\) at \(25 °C\) and \(V = 255 mL\)

Convert Celsius to Kelvin

Convert \(25 °C\) to Kelvin: \[25 + 273.15 = 298.15 K\]

Use Charles's Law to find missing temperature

Apply Charles's Law for this data set: \[\frac{247}{298.15} = \frac{255}{T_2}\] Solving for \(T_2\): \[T_2 = \frac{255 \times 298.15}{247} \] \[T_2 = 308.88 K\] The missing temperature is \(308.88 K\) in Kelvin. c. Find the missing volume for \(V = 1.00 mL\) at \(2272 °C\) and \(25 °C\)

Convert Celsius to Kelvin

Convert both temperatures to Kelvin: \[2272 + 273.15 = 2545.15 K\] \[25 + 273.15 = 298.15 K \]

Use Charles's Law to find missing volume

Now, we can apply Charles's Law for these values: \[\frac{1.00}{2545.15} = \frac{V_2}{298.15}\] To find \(V_2\): \[V_2 = \frac{1.00 \times 298.15}{2545.15}\] \[V_2 = 0.117 mL\] The missing volume is \(0.117 mL\).

Key concepts

Volume-Temperature Relationship

The volume-temperature relationship is a fundamental concept in understanding how gases behave. This relationship is governed by Charles's Law, which states that the volume of a gas is directly proportional to its temperature when pressure and number of moles of gas are held constant. In simpler terms, if you increase the temperature of a gas, its volume will also increase, provided the pressure does not change.
Charles's Law can be expressed with the formula:

• \[ V_1 \times T_2 = V_2 \times T_1 \]

This formula can be rearranged to solve for any missing value among the variables: initial volume \(V_1\), initial temperature \(T_1\), final volume \(V_2\), or final temperature \(T_2\).
This law helps us understand why balloons expand when they are heated or why car tires may burst if the temperature outside becomes very high.

Gas Laws

Gas Laws are a set of laws that describe the behavior of gases in different conditions of temperature, volume, and pressure. Apart from Charles's Law, other important gas laws include Boyle's Law and Avogadro's Law, forming a basis for the Ideal Gas Law.
Each law makes specific assumptions and predictions:

• **Boyle's Law**: At a constant temperature, the pressure of a gas is inversely proportional to its volume.
• **Avogadro's Law**: At constant pressure and temperature, the volume of a gas is directly proportional to the number of moles.
• **Ideal Gas Law**: Combines all three laws and is often written as \(PV = nRT\), where \(P\) is pressure, \(V\) is volume, \(n\) is number of moles, \(R\) is the gas constant, and \(T\) is temperature in Kelvin.

Each of these laws gives us a glimpse into the predictable nature of gases. They allow us to calculate unknown variables and understand more about everyday phenomena, such as why your basketball may feel flat on cold days.

Temperature Conversion

Temperature conversion is an essential skill when working with gas laws. Commonly, temperature needs to be in Kelvin instead of Celsius for these calculations.
To convert Celsius to Kelvin:

• \[ T(K) = T(°C) + 273.15 \]

The Kelvin scale is vital because it starts at absolute zero, which is the lowest possible temperature where particles have minimal kinetic energy. Using Kelvin prevents negative temperatures in calculations.
Without converting to Kelvin, applying gas laws can give incorrect results, since the proportionality assumed by these laws fits the absolute temperature scale.