Question

A high-altitude balloon is filled with \(1.41 \times 10^{4} \mathrm{~L}\) of hydrogen at a temperature of \(21^{\circ} \mathrm{C}\) and a pressure of 745 torr. What is the volume of the balloon at a height of \(20 \mathrm{~km}\), where the temperature is \(-48^{\circ} \mathrm{C}\) and the pressure is \(63.1\) torr? (a) \(1.274 \times 10^{5} \mathrm{~L}\) (b) \(1.66 \times 10^{5} \mathrm{~L}\) (c) \(1.66 \times 10^{4} \mathrm{~L}\) (d) None of these

Short answer

1.66 \(\times\) 10^5 L

Step-by-step solution

Convert temperatures to Kelvin

Convert the given temperatures from degrees Celsius to Kelvin, because all gas law calculations require temperatures in Kelvin. To convert from Celsius to Kelvin, add 273.15 to the Celsius temperature. For the initial temperature: T1 = 21°C + 273.15 = 294.15 K. For the final temperature: T2 = -48°C + 273.15 = 225.15 K.

Convert pressures to atm

Convert the given pressures from torr to atmospheres (atm), because calculations commonly use atm. Use the conversion factor 1 atm = 760 torr. For the initial pressure: P1 = 745 torr * (1 atm / 760 torr) = 0.9803 atm. For the final pressure: P2 = 63.1 torr * (1 atm / 760 torr) = 0.08303 atm.

Use Combined Gas Law

Apply the combined gas law which relates the pressure, volume, and temperature of a gas, keeping the number of moles constant. The combined gas law is PV/T = constant, or P1V1/T1 = P2V2/T2. Solve for the final volume V2: V2 = (P1V1T2) / (P2T1).

Insert Values and Calculate

Insert the known values into the rearranged combined gas law and solve for V2: V2 = (0.9803 atm * 1.41 \(\times\) 10^4 L * 225.15 K) / (0.08303 atm * 294.15 K) = 1.66 \(\times\) 10^5 L.

Key concepts

Gas laws in chemistry

Understanding the behavior of gases is fundamental in chemistry, and 'Gas laws in chemistry' provide the relationships between pressure, volume, temperature, and the number of particles in a gas. These laws are vital for predicting how gases will respond to changing conditions. The Combined Gas Law is one of several important gas laws and is a combination of three simpler laws - Boyle's, Charles', and Gay-Lussac's Laws. It describes the relationship between pressure (P), volume (V), and temperature (T) of a fixed amount of gas when it's subjected to changes in conditions.

If we isolate one of the variables, we can see how it impacts the other two. For instance, Boyle's Law tells us that at constant temperature, the pressure of a gas is inversely proportional to its volume. This means if you decrease the volume of a gas, the pressure will increase, as long as the temperature remains the same. Charles' Law states that volume and temperature are directly proportional, at constant pressure, signaling that if you heat a gas, its volume will increase if the pressure is held steady. Finally, Gay-Lussac's Law ties pressure and temperature together, implying that they are directly proportional when volume is held constant.

The Combined Gas Law uses these principles and allows us to solve for unknowns when a gas undergoes a change in two or more conditions, like a balloon rising to high altitudes where both temperature and pressure drop. In the context of our exercise, we apply this law to determine what happens to the volume of a hydrogen-filled balloon when it ascends from the Earth's surface to a significant height in the atmosphere.

Thermodynamics

Thermodynamics is the branch of physics that deals with the relationships between heat and other forms of energy. In the context of 'Gas laws in chemistry', it's crucial because it helps us understand how temperature affects the behavior of gases. Four laws of thermodynamics lay the groundwork for energy transfer and the work done by or on a system, like a gas inside a balloon.

The first law, also known as the law of energy conservation, states that energy cannot be created or destroyed in an isolated system. The second law, at its simplest, indicates that in any energy transfer or transformation, entropy, or disorder, in the universe will increase. If we apply these laws to the scenario of our high-altitude balloon, we could say the energy (such as heat) absorbed by the gas at lower altitudes is redistributed as the balloon ascends and the temperature drops, resulting in an increase in volume as calculated by the Combined Gas Law.

Understanding thermodynamics allows us to make predictions about the behavior of gases when they experience changes in temperature and pressure, which in turn gives us a better grasp of the real-life applications of gas laws, such as predicting the weather, designing engines, and even baking!

Chemical calculations

Chemical calculations are an essential aspect of chemistry that involves quantifying changes in chemical substances. When dealing with gases, calculations require an understanding of the mole concept, molar volume, and ideal gas law, along with the other gas laws.

Chemical calculations often involve converting units, such as temperature from Celsius to Kelvin or pressure from torr to atmospheres, to use them in equations like the Combined Gas Law. Getting these conversions right is crucial because using incorrect units can lead to wrong answers. For example, in our balloon problem, correctly converting the given temperatures to Kelvin and the pressures to atmospheres allowed us to apply the Combined Gas Law accurately and solve for the unknown volume.

In more complex chemical reactions, stoichiometry comes into play, which involves using balanced chemical equations to calculate the relative quantities of reactants and products involved in a reaction. In the context of gas laws, stoichiometry could help us determine how much of a gas is produced or consumed during a reaction under specific conditions. This highlights the importance of precise chemical calculations in understanding and predicting the behavior of gases in various scenarios.