Question

You are doing research on planet \(\mathrm{X}\). The temperature inside the space station is a carefully controlled \(24^{\circ} \mathrm{C}\) and the pressure is \(755 \mathrm{mmHg}\). Suppose that a balloon, which has a volume of \(850 . \mathrm{mL}\) inside the space station, is placed into the airlock, and floats out to planet \(X\). If planet \(X\) has an atmospheric pressure of \(0.150 \mathrm{~atm}\) and the volume of the balloon changes to \(3.22 \mathrm{~L}\), what is the temperature \(\left({ }^{\circ} \mathrm{C}\right)\) on planet \(\mathrm{X}\) ( \(n\) remains constant)?

Short answer

-103.34 °C

Step-by-step solution

Understand the Given Data

Identify the given conditions inside the space station and on planet X. Inside the space station, the temperature (T1) is 24 °C, pressure (P1) is 755 mmHg, and volume (V1) is 850 mL. On planet X, the pressure (P2) is 0.150 atm, and the volume (V2) of the balloon is 3.22 L.

Convert Units if Necessary

Convert the pressure inside the space station from mmHg to atm. Note that 1 atm = 760 mmHg, hence\(P1 = \frac{755 \text{ mmHg}}{760 \text{ mmHg/atm}} = 0.9934 \text{ atm} \). Also, convert the initial volume from mL to L, noting that 1 L = 1000 mL, hence\(V1 = \frac{850 \text{ mL}}{1000} = 0.850 \text{ L} \).

Apply the Combined Gas Law

Use the combined gas law: \[ \frac{P1 \cdot V1}{T1} = \frac{P2 \cdot V2}{T2} \].First, convert the initial temperature T1 from Celsius to Kelvin using the formula: \(T1 = 24 + 273.15 = 297.15 \text{ K} \).

Rearrange the Combined Gas Law

Solve for the final temperature T2: \[ T2 = \frac{P2 \cdot V2 \cdot T1}{P1 \cdot V1} \].

Substitute Values into the Equation

Substitute the known values into the equation: \[ T2 = \frac{(0.150 \text{ atm}) \cdot (3.22 \text{ L}) \cdot (297.15 \text{ K})}{(0.9934 \text{ atm}) \cdot (0.850 \text{ L})} \].

Calculate the Temperature

Perform the calculation: \[ T2 = \frac{0.150 \times 3.22 \times 297.15}{0.9934 \times 0.850} \approx 169.81 \text{ K} \]. Convert the temperature back to Celsius: \(T2 = 169.81 - 273.15 \approx -103.34 \^{\circ} \text{C} \).

Key concepts

Combined Gas Law

The Combined Gas Law is a relationship between pressure, volume, and temperature of a gas where the amount of gas (in moles) remains constant. It combines Charles’s Law, Boyle's Law, and Gay-Lussac's Law into a single equation: \[\frac{P1 \times V1}{T1} = \frac{P2 \times V2}{T2}\]. This means that if you know the initial and final conditions (pressure, volume, and temperature) of a gas, you can predict its behavior when one of these conditions changes. For example, in our exercise, we used the Combined Gas Law to find the final temperature (T2) on planet X when given the initial conditions inside the space station and the final volume and pressure on the planet.

Unit conversion in chemistry

Unit conversion is critical in chemistry to ensure accuracy and consistency of results. Different units are often used depending on the context. Here, we had to convert:

• Pressure from mmHg to atm, using the conversion factor that 1 atm = 760 mmHg.
• Volume from mL to L, noting that 1 L = 1000 mL.
• Temperature from Celsius to Kelvin, knowing that 0°C is equivalent to 273.15 K.

The conversions needed for this problem were:

• Pressure: 755 mmHg to 0.9934 atm.
• Volume: 850 mL to 0.850 L
• Temperature: 24 °C to 297.15 K.

Accurate conversion ensures all units are compatible for calculations, especially when applying gas laws.

Temperature and pressure relationship in gases

In gases, temperature and pressure have a direct relationship. This means if the volume is kept constant, an increase in temperature will result in an increase in pressure and vice versa.

• Boyle’s Law states that pressure is inversely proportional to volume at constant temperature: \[ P1 \times V1 = P2 \times V2. \]
• Charles's Law states that volume is directly proportional to temperature at constant pressure: \[ \frac{V1}{T1} = \frac{V2}{T2}. \]
• Gay-Lussac's Law states that pressure is directly proportional to temperature at constant volume: \[ \frac{P1}{T1} = \frac{P2}{T2}. \]

Combining these laws, the Combined Gas Law provides a comprehensive relationship where pressure, volume, and temperature can vary simultaneously. This relationship was critical in solving for the temperature on planet X based on known pressure and volume changes.

Ideal gas law

The Ideal Gas Law is an extension of the Combined Gas Law that also includes the amount of gas (in moles): \[ PV = nRT, \] where

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

The Ideal Gas Law assumes ideal behavior of gases, where gases occupy negligible space and have no interactions between molecules. In real-life scenarios, especially under high pressure and low temperature, gases deviate from ideal behavior. However, under normal conditions, the Ideal Gas Law provides a very good approximation. In the exercise, while we did not directly use the Ideal Gas Law, its concept underpins the calculations within the Combined Gas Law, particularly as we treated the gases as ideal to simplify solving for temperature.