Question

For a fixed amount of gas held at constant pressure, calculate the new volume the gas would occupy if the temperature were changed as shown in the following table. $$ \begin{array}{|c|c|c|c|} \hline \begin{array}{c} \text { Initial } \\ \text { Volume } \end{array} & \begin{array}{c} \text { Initial } \\ \text { Temperature } \end{array} & \begin{array}{c} \text { Final } \\ \text { Temperature } \end{array} & \begin{array}{c} \text { Final } \\ \text { Volume } \end{array} \\ \hline 224 \mathrm{~L} & 0.0^{\circ} \mathrm{C} & 100.0^{\circ} \mathrm{C} & ? \\\ \hline 152 \mathrm{~mL} & 45 \mathrm{~K} & 450 \mathrm{~K} & ? \\ \hline 156 \mathrm{~mL} & 45^{\circ} \mathrm{C} & 450^{\circ} \mathrm{C} & ? \\\ \hline \end{array} $$

Short answer

The final volumes for the gases are respectively: 307.1L, 1520 mL, and 353.5 mL.

Step-by-step solution

Convert Temperatures to Kelvin

Convert temperatures from Celsius to Kelvin. The formula to convert is given by \(K = °C + 273\). So, for the first row where temperature is given in Celsius, the temperatures in Kelvin will be \(0°C + 273 = 273K\) and \(100°C + 273 = 373K\).

Apply Charles's Law

The formula for Charles’s Law is \(V1/T1 = V2/T2\). Rearrange the equation to solve for the final volume \(V2\), the rearranged equation will be \(V2 = V1 * T2 / T1\).

Calculate Final Volume for First Row

Substitute the values from the first row into the rearranged Charles's law equation: \(V2 = 224L * 373K / 273K = 307.1L \).

Repeat Steps for Other Rows

Repeat step 2 for other rows, taking care to convert Celsius temperatures to Kelvin before performing operations. For the second row, since the temperatures are already in Kelvin, we can substitute directly: \(V2 = 152mL * 450K / 45K = 1520mL \). For the third row, first convert the temperatures to Kelvin: \(45°C + 273 = 318K\) and \(450°C + 273 = 723K\). Then substitute the temperatures and initial volume in the formula to find the final volume: \( V2 = 156 mL * 723K / 318K = 353.5 mL \).

Key concepts

Temperature Conversion

When working with gas laws, temperature is often a crucial factor that must be expressed in Kelvin. This is important because Kelvin is the absolute temperature scale used in scientific calculations, eliminating the possibility of negative values which can complicate calculations. If you're given a temperature in Celsius, the conversion to Kelvin is quite simple. Just add 273 to the Celsius temperature to get the Kelvin temperature. This conversion ensures consistent results across calculations as Kelvin relates directly to kinetic molecular energies. For example, if a temperature change occurs from 0°C to 100°C, converting both temperatures gives 273 K to 373 K. Doing this conversion allows you to apply gas laws like Charles's Law accurately.

Ideal Gas Law

The ideal gas law is a foundation of thermodynamics and relates the pressure, volume, and temperature of a gas with the amount of gas present. The formula for the ideal gas law is given as \[ PV = nRT \]where:

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

Although Charles's Law specifically looks at the relationship between volume and temperature at constant pressure and amount of gas, it stems from the ideal gas law. Charles's Law is actually a specific case of the ideal gas law where the pressure and moles \( n \) are held constant, simplifying it to \[ V \propto T \] meaning volume is directly proportional to temperature.

Volume and Temperature Relationship

Charles's Law defines the direct relationship between volume and temperature of a gas, showing that if the temperature of a gas increases at constant pressure, its volume increases, and inversely, if the temperature decreases, the volume decreases as well. This can be mathematically expressed as \[ \frac{V_1}{T_1} = \frac{V_2}{T_2} \] This equation implies that the ratio of the initial volume \( V_1 \) and initial temperature \( T_1 \) of the gas is equal to the ratio of its final volume \( V_2 \) and final temperature \( T_2 \). Therefore, to find the final volume after a temperature change, you rearrange the equation to: \[ V_2 = V_1 \times \frac{T_2}{T_1} \] This equation helps you solve for unknowns in situations where you know the initial conditions and one final condition. By maintaining a steady pressure, the relationship becomes straightforward since changes in temperature directly determine changes in volume.