Question

At a deep-sea station that is 200 m below the surface of the Pacific Ocean, workers live in a highly pressurized environment. How many liters of gas at STP must be compressed on the surface to fill the underwater environment with \(2.00 \times 10^{7} L\) of gas at 20.0 atm? Assume that temperature remains constant.

Short answer

4.00 × 10^8 L of gas at STP must be compressed.

Step-by-step solution

Understand the Given Information

The problem states that there is a volume of gas, 20,000,000 liters, under a pressure of 20.0 atm. We need to find the volume of this gas when it is at Standard Temperature and Pressure (STP), which is 1 atm.

Use Boyle’s Law

To solve this problem, use Boyle’s Law which states that for a given mass of ideal gas at constant temperature, the product of pressure and volume is constant. This can be written as: \[ P_1 V_1 = P_2 V_2 \]Here, \[ P_1 = 20.0 \text{ atm} \]\[ V_1 = 2.00 \times 10^{7} \text{ L} \]\[ P_2 = 1 \text{ atm} \]We need to find \[ V_2 \].

Rearrange Boyle’s Law to Solve for \(V_2\)

Rearrange the Boyle’s Law equation to solve for the unknown volume \(V_2\): \[ V_2 = \frac{P_1 V_1}{P_2} \]

Substitute the Values into the Equation

Substitute the known values into the equation: \[ V_2 = \frac{20.0 \text{ atm} \times 2.00 \times 10^{7} \text{ L}}{1 \text{ atm}} \]

Calculate

Perform the calculation: \[ V_2 = 40.0 \times 10^{7} \text{ L} \]\[ V_2 = 4.00 \times 10^{8} \text{ L} \]

Key concepts

ideal gas law

The ideal gas law is a fundamental equation in chemistry and physics that describes the behavior of ideal gases. It combines several gas laws, including Boyle's Law, Charles's Law, and Avogadro's Law. The equation is given as follows: \[ PV = nRT \]where

• P stands for pressure
• V is for volume
• n represents the number of moles of the gas
• R is the ideal gas constant
• T stands for temperature in Kelvin

Using the ideal gas law allows us to understand and predict how gases will behave under different conditions. It's especially useful when dealing with gases under a wide range of temperatures and pressures.

pressure-volume relationship

The pressure-volume relationship of gases is explained by Boyle's Law. It states that for a fixed amount of gas at constant temperature, the volume of the gas is inversely proportional to its pressure. This means if pressure increases, volume decreases and vice versa. Mathematically, it is expressed as: \[ P_1V_1 = P_2V_2 \]In this formula,

• P₁ is the initial pressure.
• V₁ is the initial volume.
• P₂ is the final pressure.
• V₂ is the final volume.

Understanding this relationship allows us to solve problems related to changes in pressure and volume, such as the deep-sea gas compression problem. In this case, once the gas is compressed at a higher pressure, its volume decreases proportionally.

standard temperature and pressure (STP)

Standard Temperature and Pressure (STP) is a standard set of conditions for measuring gases. It makes it easier to compare measurements as it provides a common scale. STP is defined as:

• Temperature: 0°C or 273.15 K
• Pressure: 1 atmosphere (atm)

At STP, one mole of an ideal gas occupies 22.4 liters. This standard condition is widely used in chemistry and physics when performing calculations involving gases. It simplifies the process of understanding how gases behave because we can always relate them back to these baseline conditions.