Question

What will be the effect on the volume of an ideal gas if the pressure is doubled and the absolute temperature is halved?

Short answer

The volume of the ideal gas will decrease to one-fourth of its initial value when the pressure is doubled and the absolute temperature is halved.

Step-by-step solution

Remember the Ideal Gas Law equation

The Ideal Gas Law equation is given by: \(PV = nRT\), where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant, and T is the absolute temperature.

Analyze the given conditions

We are given that the pressure is doubled and the absolute temperature is halved. Using P' to denote the new pressure and T' to denote the new temperature, we have: \(P' = 2P\) \(T' = \frac{1}{2}T\)

Determine the initial and final gas law equations

Using the initial pressure P, volume V, and temperature T, the initial gas law equation is: \(PV = nRT\) For the new pressure P' and temperature T', but the same amount of gas n, we have: \(P'V' = nR( T' )\)

Insert the new pressure and temperature values into the second equation

Replace P' with 2P and T' with (1/2)T in the second equation. This gives us: \((2P)V' = nR(\frac{1}{2}T)\)

Simplify the equation

Simplify the equation and solve for V': \(2PV' = \frac{1}{2} nRT\) Now notice that the term \(PV = nRT\), so we can replace \(nRT\) with \(PV\) in the equation above: \(2PV' = \frac{1}{2} PV\)

Solve for the new volume, V'

Divide both sides of the equation by 2P to find the new volume, V': \(V' = \frac{1}{4} V\)

Interpret the result

The new volume, V', is one-quarter of the initial volume, V. This means that if the pressure is doubled and the absolute temperature is halved, the volume of the ideal gas will decrease to one-fourth of its initial value.

Key concepts

Volume-Pressure Relationship

The relationship between volume and pressure is a fundamental concept in the study of gases. When we consider the volume-pressure relationship, we are often referring to Boyle's Law.
Boyle's Law states that at a constant temperature, the volume of a given amount of gas is inversely proportional to its pressure. This means that if the pressure increases, the volume decreases, and vice versa.

For example, consider a balloon. When you squeeze the balloon (increasing pressure), the volume decreases. When you let the balloon expand (decreasing pressure), the volume increases. This inverse relationship can be expressed mathematically as:

• \( P \times V = \text{constant} \)

In our original exercise, when the pressure is doubled, you'll need to look at how it affects volume. By maintaining an inverse relationship, you'll see that the volume of an ideal gas will reduce in response to increased pressure, provided temperature and other conditions remain constant.

Temperature Effect on Gases

Temperature plays a crucial role in the behavior of gases. As per Charles's Law, at constant pressure, the volume of a given mass of an ideal gas is directly proportional to its absolute temperature. In even simpler terms:

As temperature increases, so does the volume. If a gas is heated, it expands because increased temperature causes gas particles to move faster and farther apart. Conversely, when cooled, the gas particles slow down, and the volume decreases.

When dealing with Kelvin for temperature measurement, the direct relationship is expressed as:

• \( \frac{V}{T} = \text{constant} \)

The exercise presented an interesting situation where the absolute temperature was halved. This means that if pressure remains constant, the volume would decrease by half. However, sourced from the Ideal Gas Law, other factors like changes in pressure also affect the final outcome, making it crucial to analyze both adjustments together.

Gas Equations

Gas laws are foundational formulas that describe the behavior of ideal gases. They include Boyle's Law, Charles's Law, and Avogadro's Law, but the Ideal Gas Law, represented by:

• \( PV = nRT \)

This formula combines all simple gas laws into one universal equation that relates pressure (P), volume (V), and temperature (T) with the number of moles (n) and the ideal gas constant (R).

In the exercise context, we first see how pressure and temperature shifts are combined into the Ideal Gas Law to derive new states of volume. By substituting new values for pressure and temperature, we essentially harmonize these changes with:- Increase the pressure: \( P' = 2P \)- Reduce the temperature: \( T' = \frac{1}{2}T \)
This solution demonstrates that the volume of gas will adjust in response to these variables. Applying these changes, we see the resultant fraction of the initial volume, showcasing the law’s profound utility across varied conditions.