Problem 11 - University Physics with Modern Physics

Question

The gas inside a balloon will always have a pressure nearly equal to atnospheric pressure, since that is the pressure applied to the outside of the balloon. You fill a balloon with helium (a nearly ideal gas to a volume of 0.600 \(\mathrm{L}\) at a tenperature of \(19.0^{\circ} \mathrm{C} .\) What is the volume of the balloon if you cool it to the boiling point of liguid nitrogen \((77.3 \mathrm{K}) ?\)

Step-by-Step Solution

Answer

The volume of the balloon is approximately 0.1588 L.

Step 1: Understand the Problem

The problem involves a balloon with helium gas cooled to a specific temperature. We need to find out how the volume will change when the temperature is decreased from \(19.0^{\circ} \mathrm{C}\) to the boiling point of liquid nitrogen, \(77.3 \mathrm{K}\).

Step 2: Identify Relevant Concepts

This problem involves the Ideal Gas Law and Charles's Law, which states that for a given amount of gas at constant pressure, the volume of the gas is directly proportional to its temperature in Kelvin.

Step 3: Convert Celsius to Kelvin

Convert the initial temperature from Celsius to Kelvin. Kelvin is the standard unit used in gas law calculations.\[ T_1 = 19.0 + 273.15 = 292.15 \, \mathrm{K} \]

Step 4: Apply Charles's Law

Charles's Law is given by \( \frac{V_1}{T_1} = \frac{V_2}{T_2} \), where \(V_1\) and \(T_1\) are the initial volume and temperature, and \(V_2\) and \(T_2\) are the final volume and temperature. Rearrange the formula to solve for the final volume, \(V_2\):\[ V_2 = V_1 \times \frac{T_2}{T_1} \]

Step 5: Substitute Known Values

Substitute the known values into the equation:\[ V_2 = 0.600 \, \mathrm{L} \times \frac{77.3 \, \mathrm{K}}{292.15 \, \mathrm{K}} \]

Step 6: Calculate the Final Volume

Perform the calculation:\[ V_2 = 0.600 \, \mathrm{L} \times 0.2646 \approx 0.1588 \, \mathrm{L} \]

Step 7: Result

Therefore, the final volume of the balloon is approximately \(0.1588 \, \mathrm{L}\).

Key Concepts

Ideal Gas Law

The Ideal Gas Law is a powerful equation used to relate the pressure, volume, temperature, and amount of a gas. Its formula is given by:\[ PV = nRT \]where,

• P is the pressure of the gas.
• V is the volume of the gas.
• n is the amount of gas in moles.
• R is the ideal gas constant.
• T is the temperature of the gas in Kelvin.

The Ideal Gas Law helps predict how a gas behaves under various conditions. It is particularly relevant when assessing changes in volume and temperature without altering the quantity of the gas particles involved. In the context of our problem, this law supports the understanding that as temperature decreases, the volume tends to decrease as well, given a constant pressure.

temperature conversion

When performing gas law calculations, it's crucial to use the Kelvin scale for temperature. This is due to Kelvin being the absolute temperature scale, where zero reflects the absence of thermal energy. To convert from Celsius to Kelvin, add 273.15 to the Celsius temperature. This transformation is necessary to ensure that the gas laws operate correctly, given they require absolute temperatures for accurate predictions.

For example, in the provided exercise, the initial temperature is given as 19.0°C. To convert this to Kelvin, you add:\[ 19.0 + 273.15 = 292.15\, \mathrm{K} \]Converting temperatures to Kelvin eliminates risks of mathematical errors related to negative temperatures, ensuring a straightforward application of Charles's Law.

helium gas

Helium is a lightweight, inert gas that is frequently used in balloons due to its low density and non-reactive nature. It is considered nearly ideal because it adheres closely to the behavior predicted by the Ideal Gas Law. This means its molecules have minimal interactions with each other, and they occupy negligible space compared to the volume of the container they are in.

Given its simplicity and predictable behavior, helium serves as an excellent example when learning about gas laws. In experiments and calculations, helium's behavior will often align with theoretical predictions, allowing students to accurately apply equations such as Charles's Law to determine changes in volume with temperature.

liquid nitrogen boiling point

Liquid nitrogen boils at a strikingly low temperature of 77.3 Kelvin (-195.85°C). This cryogenic boiling point is instrumental in numerous scientific and commercial applications, especially those requiring severe temperature conditions, like cryopreservation and material testing.

When helium gas in a balloon is cooled to this temperature, it provides a practical example of Charles's Law. According to this law, a gas will decrease in volume as its temperature drops at constant pressure. Thus, understanding the boiling point of liquid nitrogen allows students to visualize and comprehend how drastic temperature changes can affect gas volume. In this way, liquid nitrogen serves as a critical reference point in learning temperature-related gas behaviors.