Question

A balloon contains 146.0 mL of gas confined at a pressure of 1.30 atm and a temperature of 5.0ºC. If the pressure doubles and the temperature decreases to 2.0ºC, what will be the volume of gas in the balloon?

Short answer

The volume of the gas in the balloon after the pressure doubles and the temperature decreases to 2.0ºC is approximately \(142.6\,\text{mL}\).

Step-by-step solution

Understand the problem and write initial and final conditions of the gas

First, let's write down the initial and final conditions of the gas: Initial state: Volume = \(V_1 = 146.0\,\text{mL}\) Pressure = \(P_1 = 1.30\,\text{atm}\) Temperature = \(T_1 = 5.0\,^\circ\text{C}\) Final state: Volume = \(V_2=?\) Pressure = \(P_2 = 2P_1 = 2(1.30)\,\text{atm}\) Temperature = \(T_2 = 2.0\,^\circ\text{C}\) We need to find the volume of the gas in its final state, \(V_2\).

Convert the temperatures to Kelvin

Since the Combined Gas Law requires the temperatures to be in Kelvin, we need to convert the Celsius temperatures to Kelvin. To do this, we can use the following formula: \(T(K) = T(^\circ\text{C}) + 273.15\) Initial temperature in Kelvin: \(T_1 = 5.0\,^\circ\text{C} + 273.15 = 278.15\,\text{K}\) Final temperature in Kelvin: \(T_2 = 2.0\,^\circ\text{C} + 273.15 = 275.15\,\text{K}\) Now we have the temperatures in Kelvin: \(T_1 = 278.15\,\text{K}\) \(T_2 = 275.15\,\text{K}\)

Apply Combined Gas Law formula

The Combined Gas Law formula is given by: \(\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}\) Now we can substitute the values of the initial and final pressure, volume, and temperatures in this formula: \(\frac{1.30\,\text{atm} \times 146.0\,\text{mL}}{278.15\,\text{K}} = \frac{(2\times1.30)\,\text{atm} \times V_2}{275.15\,\text{K}}\)

Solve for \(V_2\)

To solve for \(V_2\), we will first simplify the equation: \(V_2 = \frac{(1.30\,\text{atm} \times 146.0\,\text{mL})\times 275.15\,\text{K}}{(2\times 1.30) \,\text{atm} \times 278.15\,\text{K}}\) Then calculate the value of \(V_2\): \(V_2 \approx 142.6\,\text{mL}\)

State the final answer

The volume of the gas in the balloon after the pressure doubles and the temperature decreases to 2.0ºC is approximately 142.6 mL.

Key concepts

Understanding Temperature Conversion

In the Combined Gas Law, temperature is an essential component because it affects the state of gases. Typically, temperatures must be expressed in Kelvin for gas law calculations. This conversion from Celsius to Kelvin ensures that calculations remain accurate, as Kelvin begins at absolute zero, the theoretical point where particles cease to move. To convert Celsius to Kelvin, simply add 273.15 to the Celsius temperature. For example, if you have a temperature of 5.0°C, its conversion to Kelvin would work like this:

• Convert using \(T(K) = T(\, ^\circ\text{C}) + 273.15 \)
• 5.0°C becomes \(5.0 + 273.15 = 278.15\, \text{K}\)

When tackling problems involving gas laws, always remember to first convert temperatures to Kelvin before substituting them into any formulas. This simple step can prevent potentially large errors down the line.

Gas Volume Calculation Using the Combined Gas Law

Calculating gas volume under changing conditions requires a solid understanding of the Combined Gas Law. This law relates the pressure, volume, and temperature of a gas by showing how they interact. The formula is represented as \[\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}\]where:

• \(P_1\), \(V_1\), and \(T_1\) are the initial pressure, volume, and temperature.
• \(P_2\), \(V_2\), and \(T_2\) are the final pressure, volume, and temperature.

This formula allows us to solve for any unknown variable when the other five are known. For example, to find a new volume \(V_2\), rearrange the formula to: \[V_2 = \frac{P_1 V_1 T_2}{P_2 T_1}\]Using this rearranged equation, you can calculate how changes in pressure and temperature affect gas volume, such as in our exercise where the gas expands or contracts based on these parameters. Remember, while performing these calculations, ensure all your temperatures are converted to Kelvin and pressures are consistently measured.

Pressure and Temperature Relationship in Gases

The relationship between pressure and temperature within the context of gases is crucial to understanding the behavior of gases under various conditions. According to the Combined Gas Law, temperature changes can lead directly to changes in pressure, assuming the volume of the gas is constant. Here's why this happens:

• As temperature increases, the kinetic energy of gas particles increases. This causes particles to collide more forcefully with the walls of their container, resulting in higher pressure.
• Conversely, a decrease in temperature reduces particle movement, leading to decreased pressure.

In our example, when the temperature of the balloon's gas drops, the pressure initially at 1.30 atm also contributes to a predicted decrease in the volume, making it crucial to balance these changes to predict the balloon's capacity accurately. Understanding this natural relationship between pressure and temperature can allow for predicting changes in gas behavior, a principle used in various industrial and scientific applications.