Question

To raise the volume of a gas by four times the following method may be adopted. Which of the method is wrong? (1) \(T\) is doubled and \(P\) is also doubled (2) Keeping \(P\) constant, \(T\) is raised by two times (3) Temperature is doubled and pressure is halved (4) Keeping temperature constant pressure is reduced to \(1 / 4\) of its initial value

Short answer

Method 1 is wrong.

Step-by-step solution

Understand the Ideal Gas Law

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

Method 1 Analysis

In Method 1, both \(T\) and \(P\) are doubled. Substitute into the Ideal Gas Law: \[ P_2 = 2P \] \[ T_2 = 2T \] \[ P_2V_2 = nR T_2 \rightarrow 2P V_2 = nR(2T) \rightarrow V_2 = V \] Since \(V_2 = V\), Method 1 does not change the volume, this method is incorrect.

Method 2 Analysis

In Method 2, \(P\) is constant and \(T\) is doubled. Substitute into the Ideal Gas Law: \[ T_2 = 2T \] \[ PV_2 = nR T_2 \rightarrow P V_2 = nR(2T) \rightarrow V_2 = 2V \] Thus, the volume doubles, but we were asked to make the volume four times, so this method is also not adequate but not necessarily completely wrong.

Method 3 Analysis

In Method 3, \(T\) is doubled and \(P\) is halved. Substitute into the Ideal Gas Law: \[ T_2 = 2T \] \[ P_2 = \frac{P}{2} \] \[ P_2V_2 = nR T_2 \rightarrow \frac{P}{2} V_2 = nR(2T) \rightarrow V_2 = 4V \] Thus, the volume increases four times, so this method is correct.

Method 4 Analysis

In Method 4, \(T\) is constant and \(P\) is reduced to \(\frac{1}{4}\) of its initial value. Substitute into the Ideal Gas Law: \[ P_2 = \frac{P}{4} \] \[ P_2V_2 = nR T \rightarrow \frac{P}{4} V_2 = nRT \rightarrow V_2 = 4V \] Thus, the volume increases four times, making this method correct.

Key concepts

gas volume

Gas volume refers to the space that gas occupies. The volume of a gas is not fixed; it changes with adjustments in pressure and temperature, as described by the Ideal Gas Law.
When dealing with gases, unlike solids and liquids, the volume is highly responsive to external conditions. For instance, if you compress a gas, you can significantly reduce its volume.
Volume is typically measured in liters (L) or cubic meters (m³) and plays a crucial role in gas-related calculations. Understanding how volume interacts with other variables such as temperature and pressure is fundamental in physics and chemistry.
Properly understanding this concept can help you analyze methods for adjusting gas volume based on given conditions.

pressure and temperature relationship

The relationship between pressure and temperature in a gas is interconnected and follows Charles's Law and Gay-Lussac's Law.
According to Charles's Law, if the pressure is kept constant, the volume of a gas is directly proportional to its temperature. That means doubling the temperature will double the volume.
Gay-Lussac's Law states that if the volume is constant, pressure is directly proportional to temperature. Therefore, increasing the temperature will increase the pressure.
The Ideal Gas Law combines these relationships, and it is expressed as: the equation: PV = nRT where P is the pressure, V is the volume, n is the number of moles, R is the gas constant, and T is the temperature in Kelvin.
This equation helps to understand how changes in temperature and pressure affect the overall behavior of gases.

gas laws in chemistry

Gas laws in chemistry are fundamental laws that describe how gases behave under different conditions. The three major gas laws are Boyle's Law, Charles's Law, and Avogadro's Law.
Boyle's Law states that the pressure and volume of a gas are inversely proportional at constant temperature: PV = constant. Charles's Law suggests that the volume of a gas is directly proportional to its temperature at constant pressure: V/T = constant. Avogadro's Law indicates that the volume of a gas is directly proportional to the number of moles of gas at constant temperature and pressure: V/n = constant. These individual laws come together in the Ideal Gas Law, which is the most comprehensive equation. Analyzing these laws can provide great insights into predicting how changes in conditions will affect gas behavior, crucial for tasks like solving the textbook exercise mentioned above. Understanding these principles is essential for mastering gas-related problems in chemistry.