Question

What pressure is required to compress 5 liters of gas at 1 atm. pressure to 1 liter at a constant temperature?

Short answer

The pressure required to compress 5 liters of gas at 1 atm to 1 liter at a constant temperature is \(5 atm\).

Step-by-step solution

List the given values and the unknown value

We are given: - Initial pressure (P1) = 1 atm - Initial volume (V1) = 5 liters - Final volume (V2) = 1 liter Unknown: - Final pressure (P2)

Apply Boyle's law

According to Boyle's law, the product of pressure and volume is constant for a given temperature. This means that P1 * V1 = P2 * V2, where P1 and V1 are the initial pressure and volume, and P2 and V2 are the final pressure and volume. We will use this formula to find the final pressure (P2).

Substitute the given values into the formula

Now, we can substitute the given values of P1, V1, and V2 into the Boyle's law formula: (1 atm) * (5 liters) = P2 * (1 liter)

Solve for P2

To isolate P2, we can divide both sides of the equation by 1 liter: P2 = (1 atm) * (5 liters) / (1 liter)

Calculate the final pressure

Performing the calculation, we get: P2 = 5 atm Therefore, the pressure required to compress 5 liters of gas at 1 atm to 1 liter at a constant temperature is 5 atm.

Key concepts

Gas Compression

Gas compression involves reducing the volume of a gas while it remains in a confined space. This process is typically accompanied by changes in the pressure of the gas. According to Boyle's Law, which governs gas behavior under compression, the relationship between the pressure and volume of a gas is inversely proportional when temperature is held constant.

In simpler terms, as you compress the gas and its volume decreases, the pressure increases, provided the temperature does not change. This is crucial for understanding how gas behaves in various applications, such as in a piston or an air compressor. The concept of gas compression is significant for many industries, from automotive to heating and refrigeration.

Pressure Calculation

Calculating the pressure of a compressed gas can be straightforward when using Boyle's Law. The law states that the product of a gas's initial pressure and volume is equal to the product of its final pressure and volume, expressed as \( P_1 \times V_1 = P_2 \times V_2 \).

This formula allows you to determine the unknown variable—whether it’s the initial pressure, final pressure, initial volume, or final volume—by rearranging the values based on which ones are known.

In our exercise, we needed to find the final pressure, so we rearranged the formula to solve for \( P_2 \), giving us \( P_2 = \frac{P_1 \times V_1}{V_2} \). By substituting the given values, we calculated the final pressure after compression.

Constant Temperature

When discussing gas laws, constant temperature is a crucial condition. In Boyle’s Law, it is assumed that the temperature remains constant during the compression or expansion of a gas. This condition is important because the behavior of a gas is also influenced by temperature changes, following Charles' Law.

If the temperature were to change, the relationship between pressure and volume would not follow Boyle’s Law. Instead, it would shift based on the altered state of the gas molecules. Thus, ensuring a constant temperature allows us to reliably predict the effects of changing pressure and volume. Constant temperature situations are termed as "isothermal processes."

Initial and Final Volume

The initial and final volumes of a gas are key factors in understanding gas compression. Volume represents the amount of space a gas occupies. In our exercise, the initial volume \( V_1 \) was 5 liters, and the final volume \( V_2 \) was 1 liter. As the volume decreased, according to Boyle's Law, the pressure increased.

Recognizing the changes in volume is vital in scenarios where volume directly influences the outcome, such as calculating the necessary pressure to achieve a specific volume. It is also essential in designing systems where precise volume adjustments are needed, ensuring efficiency and safety in operations.