Question

A compressed-air tank carried by scuba divers has a volume of \(6.80 \mathrm{~L}\) and a pressure of 120 atm at \(20^{\circ} \mathrm{C}\). What is the volume of air in the tank at \(0{ }^{\circ} \mathrm{C}\) and 1.00 atm pressure (STP)?

Short answer

The volume at STP is 907.06 L.

Step-by-step solution

Identify the Problem

We're given the initial conditions of a gas in a tank: volume \(V_1 = 6.80\,\mathrm{L}\), pressure \(P_1 = 120\,\mathrm{atm}\), and temperature \(T_1 = 20^{\circ} \mathrm{C}\). We need to find the volume \(V_2\) at standard temperature and pressure (STP), which is at \(0^{\circ}\mathrm{C}\) and \(1.00\,\mathrm{atm}\).

Convert Temperatures to Kelvin

Convert the initial and final temperatures from Celsius to Kelvin using the formula \(T(K) = T(^{\circ}\mathrm{C}) + 273.15\). Thus, \(T_1 = 20 + 273.15 = 293.15\,\mathrm{K}\) and \(T_2 = 0 + 273.15 = 273.15\,\mathrm{K}\).

Use the Ideal Gas Law

Use the combined gas law \(\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}\) to solve for \(V_2\). This law applies because the amount of gas is constant.

Substitute Known Values

Substitute the known values into the equation: \(\frac{120\,\mathrm{atm} \cdot 6.80\,\mathrm{L}}{293.15\,\mathrm{K}} = \frac{1.00\,\mathrm{atm} \cdot V_2}{273.15\,\mathrm{K}}\).

Solve for \(V_2\)

Solve the equation for \(V_2\) by multiplying both sides by \(273.15\,\mathrm{K}\) and then dividing by \(1.00\,\mathrm{atm}\). This gives \(V_2 = \frac{120\,\mathrm{atm} \cdot 6.80\,\mathrm{L} \cdot 273.15\,\mathrm{K}}{293.15\,\mathrm{K} \cdot 1.00\,\mathrm{atm}}\). Simplify to find \(V_2 = 907.06\,\mathrm{L}\).

Key concepts

Combined Gas Law

The combined gas law is a foundational principle in chemistry that helps us understand how gases behave under different conditions. This law combines three important gas laws: Boyle’s Law, Charles’s Law, and Gay-Lussac's Law. Each of these laws examines a different aspect of how volume, pressure, and temperature interact.

• Boyle’s Law: Links pressure and volume, stating that if the temperature is held constant, the pressure of a gas is inversely proportional to its volume.
• Charles’s Law: Shows the relationship between temperature and volume, where volume is directly proportional to the temperature if pressure remains constant.
• Gay-Lussac's Law: Connects temperature to pressure, stating that pressure is directly proportional to temperature when volume is fixed.

In the combined gas law, we assume the amount of gas does not change. We express this relationship in the formula \(\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}\). This allows us to calculate the unknown property of a gas when its state changes, such as finding a new volume under different temperature and pressure conditions.

Temperature Conversion

Conversion of temperature from Celsius to Kelvin is a necessary step in many scientific calculations. This is because the Kelvin scale is an absolute temperature scale that begins at zero and avoids negative values, making it ideal for scientific use.
To convert Celsius to Kelvin, simply add 273.15 to the Celsius temperature. For example:

• If the initial temperature is 20°C, then in Kelvin it is 293.15 K.
• Likewise, 0°C is equivalent to 273.15 K.

This conversion is particularly useful when working with gas laws because thematically, gases behave according to their energy, which is directly related to temperature. It's crucial to always perform calculations with temperature in Kelvin to maintain consistency and accuracy in your results.

Pressure and Volume Relationship

Understanding the relationship between pressure and volume is crucial for predicting how gases will respond to changes in their environment. This relationship is clearly described by Boyle’s Law, which states that at constant temperature, the pressure of a gas is inversely proportional to its volume.
In practical terms, when you increase the pressure exerted on a gas, its volume will decrease if the temperature remains constant. Conversely, decreasing the pressure allows the gas to expand, increasing its volume.

• This concept explains why a balloon might burst if squeezed too hard – the pressure on the gas inside increases, forcing it into a smaller space until it cannot be contained.
• Similarly, releasing some of that pressure by allowing air to escape from a cycle pump will cause the volume of air in the pump to expand.

Understanding this inverse relationship is key in industries like scuba diving, where divers need to manage air supply effectively under varying pressure conditions beneath water.