Calculate the Volume of an Ideal Gas at 1 atm Using Boyle’s Law

Correct Answer: 2500 L

Understanding the Pressure-Volume Relationship of an Ideal Gas

This question describes an ideal gas whose pressure changes while the temperature remains constant. Whenever a fixed amount of gas undergoes a change in pressure and volume at constant temperature, the relationship between pressure and volume is explained by Boyle’s law.

According to Boyle’s law, the pressure of a fixed amount of gas is inversely proportional to its volume when the temperature remains constant. This means that when the pressure decreases, the volume increases. Similarly, when the pressure increases, the volume decreases.

Mathematically, this relationship can be written as:

P ∝ 1/V

Therefore:

PV = constant

For two different states of the same gas at constant temperature, Boyle’s law is expressed as:

P₁V₁ = P₂V₂

Here, P₁ and V₁ represent the initial pressure and initial volume, while P₂ and V₂ represent the final pressure and final volume of the gas.

Given Values in the Question

The initial volume of the ideal gas is:

V₁ = 50 L

The initial pressure of the gas is:

P₁ = 50 atm

The final pressure of the gas is:

P₂ = 1 atm

The final volume V₂ is unknown and must be calculated. Since the question clearly states that the temperature remains unchanged, Boyle’s law can be applied directly.

Step-by-Step Calculation of the Final Gas Volume

Step 1: Apply Boyle’s Law

For an ideal gas at constant temperature:

P₁V₁ = P₂V₂

Substituting the given values:

50 × 50 = 1 × V₂

Step 2: Calculate the Final Volume

Multiplying the values on the left-hand side:

2500 = V₂

Therefore:

V₂ = 2500 L

The calculated value is already a whole number, so rounding it to the nearest integer does not change the result.

Why Does the Gas Volume Increase to 2500 L?

The pressure of the gas decreases from 50 atm to 1 atm. In other words, the final pressure is 50 times smaller than the initial pressure. Since pressure and volume are inversely proportional at constant temperature, the volume must become 50 times larger.

The initial volume is 50 L. Therefore:

Final volume = 50 × 50 L

Final volume = 2500 L

This large increase in volume is consistent with Boyle’s law. As the external pressure acting on the gas decreases significantly, the gas expands until its pressure reaches the new value of 1 atm.

Derivation from the Ideal Gas Equation

The same result can also be understood using the ideal gas equation:

PV = nRT

For the same amount of gas, the number of moles n remains constant. The gas constant R is also constant, and the question states that the temperature T does not change. Therefore, the product PV must remain constant.

Thus:

P₁V₁ = P₂V₂

This shows that Boyle’s law is directly obtained from the ideal gas equation for an isothermal process involving a fixed amount of gas.

Physical Meaning of an Isothermal Process

A process in which the temperature remains constant is called an isothermal process. In this question, the gas expands from a high-pressure state of 50 atm to a low-pressure state of 1 atm while maintaining the same temperature.

Because the temperature of an ideal gas is directly related to the average kinetic energy of its molecules, constant temperature means that the average molecular kinetic energy remains unchanged. The change occurs mainly in the spacing between the gas molecules as the gas expands to occupy a much larger volume.

Checking the Answer Using the Inverse Relationship

The pressure changes by the factor:

P₁/P₂ = 50/1 = 50

Therefore, the volume must change by the inverse factor:

V₂/V₁ = 50

Hence:

V₂ = 50 × 50 = 2500 L

This confirms that the calculated final volume is correct.

Final Answer

The volume of the ideal gas at a pressure of 1 atm is 2500 L.

Using Boyle’s law, P₁V₁ = P₂V₂, the decrease in pressure from 50 atm to 1 atm causes the volume to increase from 50 L to 2500 L when the temperature remains constant.