Question

Your spaceship has docked at a space station above Mars. The temperature inside the space station is a carefully controlled \(24{ }^{\circ} \mathrm{C}\) at a pressure of \(745 \mathrm{mmHg}\). A balloon with a volume of \(425 \mathrm{~mL}\) drifts into the airlock where the temperature is \(-95^{\circ} \mathrm{C}\) and the pressure is \(0.115 \mathrm{~atm}\). What is the new volume of the balloon \((n\) remains constant)? Assume that the balloon is very elastic.

Short answer

The new volume of the balloon is approximately 2148 mL.

Step-by-step solution

Convert Temperatures to Kelvin

First, convert the given temperatures from Celsius to Kelvin using the formula: \[ T(K) = T(°C) + 273.15 \]For the inside temperature: \[ 24^{\text{°C}} \rightarrow 24 + 273.15 = 297.15 \text{ K} \]For the airlock temperature: \[ -95^{\text{°C}} \rightarrow -95 + 273.15 = 178.15 \text{ K} \]

Convert Pressure Units

Next, convert the given pressures to the same unit. We'll convert the inside pressure from \( \text{mmHg} \) to \( \text{atm} \) using the conversion factor \( 1 \text{ atm} = 760 \text{ mmHg} \).For the inside pressure: \[ 745 \text{ mmHg} \rightarrow \frac{745}{760} = 0.980 \text{ atm} \]

Apply the Combined Gas Law

The combined gas law can be used to find the new volume, which is given by: \[ \frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2} \]Substitute the known values into the formula: \[ \frac{0.980 \text{ atm} \times 425 \text{ mL}}{297.15 \text{ K}} = \frac{0.115 \text{ atm} \times V_2}{178.15 \text{ K}} \]

Solve for the New Volume

Rearrange the equation to solve for \( V_2 \): \[ V_2 = \frac{0.980 \text{ atm} \times 425 \text{ mL} \times 178.15 \text{ K}}{297.15 \text{ K} \times 0.115 \text{ atm}} \]Calculate the above expression to find the new volume: \[ V_2 = \frac{74445.475}{34.66275} \text{ mL} \approx 2148 \text{ mL} \]

Key concepts

Gas Laws

Gas laws are a group of physical laws that describe the behavior of gases in various conditions. These laws help us understand the relationships between pressure (P), volume (V), and temperature (T). The most commonly used gas laws are Boyle's Law, which states that at constant temperature, the pressure of a gas is inversely proportional to its volume, and Charles's Law, which states that the volume of a gas is directly proportional to its temperature at constant pressure. These foundational concepts lead to the Combined Gas Law, which integrates both Boyle's and Charles's laws to provide a comprehensive equation:

• • Boyle's Law: For a constant temperature, the pressure and volume of a gas are inversely proportional: • Charles's Law: For a constant pressure, the volume of a gas is directly proportional to its temperature: The combined gas law is useful for understanding gas behavior in practical applications. Always keep these relationships in mind when analyzing any gas-related problem.

Ideal Gas Law

An extension of the Combined Gas Law is the Ideal Gas Law. This law incorporates the amount of gas in the system, represented by the number of moles (n). The Ideal Gas Law is expressed as: The Ideal Gas Law equation provides a comprehensive understanding by including the mole of the gas as an additional variable. It helps predict the behavior of a gas under various conditions. This equation can be used in scenarios where changes in pressure, temperature, or volume occur. Also, knowing the number of moles of the gas allows you to find any unknown parameter if the others are given. Keep in mind that ideal gas behavior assumes no intermolecular forces and that the gas particles occupy no volume. Although real gases deviate slightly from this behavior, the Ideal Gas Law provides a good approximation for most conditions.

Temperature Conversion

Temperature plays a crucial role in gas behavior. Converting temperatures from Celsius to Kelvin is often necessary, as gas laws use the Kelvin scale. Converting Celsius to Kelvin involves a simple formula: The temperature in Kelvin: The reason Kelvin is used is because it starts at absolute zero, the lowest physically possible temperature, eliminating negative values and simplifying calculations. Always remember to convert Celsius to Kelvin when working with any gas law equations to ensure accuracy.

Pressure Conversion

Pressure is another fundamental aspect of gas behavior. It can be measured in various units, such as atmospheres (atm), millimeters of mercury (mmHg), and pascals (Pa). Converting between these units is sometimes necessary to apply gas laws correctly. For instance, to convert mmHg to atm, use the conversion factor: 1 atm = 760 mmHgThe conversion factor ensures that all pressure values are in the same unit, making calculations more straightforward. Understanding how to convert pressure units is essential for accurately solving gas law problems. Always check the units given in a problem and convert if necessary to ensure consistency.