Question

You have 1 mole of an ideal gas in a balloon. How must the volume change so that the pressure in the balloon doubles and the temperature in \(\mathrm{C}\) is halved (assume the temperature is a positive value)? a. The volume must decrease by more than half the original volume. b. The volume must decrease to exactly half the original volume. c. The volume must not change because pressure and temperature "balance" out. d. The volume must increase to exactly twice the original volume. e. The volume must increase by more than twice the original volume.

Short answer

a. The volume must decrease by more than half the original volume.

Step-by-step solution

Recall the Ideal Gas Law

The Ideal Gas Law is represented by the formula: \[PV = nRT\] 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 temperature in Kelvin.

Set up two equations for the initial and final states of the gas

We will denote the initial state of the gas with P, V, and T while the final state of the gas will be denoted with P', V', and T'. We have 1 mole of the gas, and the given values are: Initial state (P, V, T) Final state (2P, V', 0.5T) Now, let's write down the Ideal Gas Law equations for both states: \(PV = nRT\) (initial state) \(2P\cdot V' = nR\cdot (0.5T)\) (final state)

Solve for the volume in the final state (V')

Since we have two equations and two unknowns (V and V'), we can solve the system of equations to find the unknown volume V': Divide the second equation by the first equation: \[\frac{2PV' }{PV} =\frac{0.5nRT}{nRT}\] Cancel out the terms that appear on both sides: \[\frac{2V'}{V} = 0.5\]

Find the relationship between the initial and final volumes

Now, we will find the relationship between the initial and final volumes: \[2V' = 0.5V\] Divide both sides by 2: \[V' = 0.25V\] This result tells us that the final volume of the gas (V') must decrease to one-quarter of the initial volume (V). Based on this, the correct answer is: a. The volume must decrease by more than half the original volume.

Key concepts

Mole Concept

The concept of a mole is crucial in understanding how quantities of substances are measured in chemistry. A mole represents a fixed number of particles, usually atoms or molecules, equal to Avogadro's number, approximately \(6.022 \times 10^{23}\). This value allows chemists to translate between mass, number of molecules, and volume in chemical reactions and processes.
In the context of the Ideal Gas Law, the number of moles \(n\) plays a vital role. It is central to calculating how a gas behaves under different conditions of pressure, volume, and temperature. In this exercise, we specifically started with 1 mole of an ideal gas. The simplicity of using one mole allows for easier calculations and underscores how a changes in state (like pressure, volume, or temperature) affect the gas.

Pressure-Volume Relationship

The pressure-volume relationship is a fundamental aspect of gas laws. According to Boyle's Law, if the temperature is held constant for a given amount of gas, the pressure of the gas is inversely proportional to its volume. This means that if you increase the pressure, the volume decreases, and vice versa.
In the Ideal Gas Law, this relationship is part of the equation \(PV = nRT\). In this problem, however, both pressure and temperature are changing. Initially, the pressure is \(P\), and the final pressure is \(2P\). The equation provided in the solution shows how these quantities interact to derive the new volume required for the gas's state to reach the desired pressure and temperature. Understanding this relationship allows you to determine how the variables will adjust if one of them changes.

Temperature Conversion

Temperature conversion is often necessary in chemistry when dealing with gases, especially when using the Ideal Gas Law. Temperatures in the Ideal Gas Law are expressed in Kelvin, which is the SI unit of temperature. To convert Celsius to Kelvin, simply add 273.15 to the Celsius temperature.
In this exercise, you need to halve the Celsius temperature, but remember, temperature transformations must be converted to Kelvin first. This ensures precision, as temperature converted incorrectly could lead to errors in solving gas law problems. The Kelvin scale removes the possibility of dealing with negative temperatures, aligning with the absolute nature of thermodynamic properties, which is essential when calculating properties like volume and pressure.

Mathematical Problem-Solving in Chemistry

Problem-solving is a critical skill in chemistry, often involving mathematical equations and logical reasoning. The solution to this problem showcases a step-by-step approach using the Ideal Gas Law.
First, each state of the gas must be expressed using the law: initial \((PV = nRT)\) and final \((2P \cdot V' = nR \cdot (0.5T))\).
The strategy involved refers to a system of equations to eliminate variables and focus on solving for the final component of interest, in this case, volume, \(V'\). Cancelling similar terms simplifies the expression, leading to a straightforward calculation for \(V'\), which must decrease to 0.25V in this exercise. Understanding this process can enhance your skills not only in gas law problems but also broadly in applied scientific calculations.