Question

A sealed balloon is filled with 1.00 \(\mathrm{L}\) helium at \(23^{\circ} \mathrm{C}\) and 1.00 atm . The balloon rises to a point in the atmosphere where the pressure is 220 . torr and the temperature is \(-31^{\circ} \mathrm{C}\) . What is the change in volume of the balloon as it ascends from 1.00 atm to a pressure of 220 . torr?

Short answer

The change in volume of the balloon as it ascends from 1.00 atm to a pressure of 220 torr is approximately \(2.774 L\).

Step-by-step solution

Convert units to appropriate values

First, we will convert temperatures from Celsius to Kelvins, and pressure from torr to atmospheres. Initial temperature: \(T_1 = (23+273) K = 296 K\) Final temperature: \(T_2 = (-31+273) K = 242 K\) Initial pressure: \(P_1 = 1.00~atm\) Final pressure: \(P_2 = \frac{220}{760}~atm\)

Apply the Ideal Gas Law

Using the Ideal Gas Law formula: \(PV = nRT\), where: - P is the pressure of the gas - V is the volume of the gas - n is the number of moles of the gas - R is the ideal gas constant (\(0.0821~L\cdot atm\cdot K^{-1}\cdot mol^{-1}\)) - T is the temperature of the gas in Kelvins Since the balloon is sealed and the amount of gas (n) remains constant, we can write the combined gas law formula as: \(\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}\) Rearranging the formula to solve for the final volume, \(V_2\): \(V_2 = \frac{P_1V_1T_2}{P_2T_1}\)

Calculate the final volume, \(V_2\)

Using the values from Step 1, plug into the equation from Step 2 to find the final volume: \(V_2 = \frac{(1.00~atm)(1.00~L)(242~K)}{(\frac{220}{760}~ atm)(296~K)}\) \(V_2 \approx 3.774 L\)

Calculate the change in volume

Now, we will calculate the change in volume as the balloon ascends: \(\Delta V = V_2 - V_1\) \(\Delta V = 3.774 L - 1.00 L = 2.774 L\) So the change in volume of the balloon as it ascends from 1.00 atm to a pressure of 220 torr is approximately 2.774 L.

Key concepts

Temperature Conversion

When dealing with gas laws, converting temperature from Celsius to Kelvin is crucial. The reason for this is straightforward: Kelvin is the absolute temperature scale used in thermodynamics. The Ideal Gas Law and its related equations require temperature to be in Kelvin to ensure accuracy and consistency.

Here’s how you perform a temperature conversion from Celsius to Kelvin:

• Identify the temperature in Celsius that needs conversion.
• Add 273 to the Celsius temperature. This shift is because 0 Kelvin is absolute zero, and it's the starting point where thermodynamic temperature begins.

In our problem, we have:

• Initial temperature: 23°C was converted to 296K. Simply, 23 + 273 = 296.
• Final temperature: -31°C was converted to 242K. -31 + 273 = 242.

This simple formula makes it easy to convert any Celsius temperature to Kelvin, helping you keep your calculations accurate.

Pressure Conversion

In this exercise, converting pressures is key to using the Ideal Gas Law appropriately. Often, chemistry problems use different units for pressure such as torr, and it’s important to convert these to atmospheres to maintain consistency in calculations, especially when using the Ideal Gas Law constant.

• To convert torr to atmospheres, use the conversion factor: 1 atm = 760 torr.
• Divide the pressure value in torr by 760 to convert it to atm.

For instance:

• The given pressure of 220 torr converts to atm by 220/760. That gives us approximately 0.289 atm.

This conversion ensures that all parameters in your equations use the appropriate units, making your calculations valid according to gas laws.

Combined Gas Law

The Combined Gas Law integrates Boyle's, Charles's, and Avogadro's laws to analyze how pressure, volume, and temperature influence a gas. It is especially relevant when dealing with problems involving changes in conditions of a gas.**Understanding the Formula:**The formula for the Combined Gas Law is:\[\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}\]Where:

• \(P_1\), \(V_1\), \(T_1\): Initial pressure, volume, and temperature.
• \(P_2\), \(V_2\), \(T_2\): Final pressure, volume, and temperature.

Use it to relate the initial and final states of the gas, especially when the amount of gas remains constant—like in our balloon problem.

In rearranging this to solve for the unknown final volume \(V_2\), we use:\[V_2 = \frac{P_1V_1T_2}{P_2T_1}\]This form is convenient for solving for any unknown when the other variables are known. It connects the conditions pre- and post-change, reflecting the gas's response to environmental changes.

Volume Change Calculation

To determine how a gas's volume changes, the last step involves computing the volume change. In our example, after applying the gas law relations and finding the final volume, we calculate the volume change using a simple subtraction.

Here's the procedure:

• Calculate the final volume \(V_2\) using the rearranged Combined Gas Law.
• Subtract the initial volume \(V_1\) from \(V_2\) to find the volume change \(\Delta V\).

For our balloon:

• Initial volume \(V_1\) was 1.00 L.
• Calculated final volume \(V_2\) was approximately 3.774 L.
• Therefore, \(\Delta V = 3.774 L - 1.00 L = 2.774 L\).

The entire process shows how gases expand or contract when subjected to different pressure and temperature conditions, as seen in the increase in the helium balloon's volume as it rises.