Problem 51 - A Complete Resource Book in Chemistry for JEE Main

Question

The density of neon will be highest at (a) STP (b) \(0^{\circ} \mathrm{C}, 2 \mathrm{~atm}\) (c) \(273^{\circ} \mathrm{C}, 1 \mathrm{~atm}\) (d) \(273^{\circ} \mathrm{C}, 2 \mathrm{~atm}\)

Step-by-Step Solution

Answer

The density of neon will be highest at condition (b): \(0^{\circ} C, 2 \text{ atm}\).

Step 1: Understanding Density in Terms of Ideal Gas Law

Density is defined as mass per unit volume. For gases, we can use the Ideal Gas Law to express density. The Ideal Gas Law is given by \( PV = nRT \), where \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the gas constant, and \( T \) is temperature in Kelvin. Density \( \rho \) can be expressed as \( \rho = \frac{PM}{RT} \), where \( M \) is the molar mass of the gas.

Step 2: Analyzing Conditions for Density

From the formula \( \rho = \frac{PM}{RT} \), density is directly proportional to pressure (\( P \)) and inversely proportional to temperature (\( T \)). This means that density increases with higher pressure and decreases with higher temperature.

Step 3: Evaluate Given Conditions

Calculate the density for each condition using the qualitative analysis from Step 2:- **(a) STP**: Standard temperature (273 K) and pressure (1 atm)- **(b) \(0^{\circ}\text{C}, 2 \text{ atm}\)**: Temperature is 273 K, pressure is 2 atm- **(c) \(273^{\circ}\text{C}, 1 \text{ atm}\)**: Temperature is 273 + 273 = 546 K, pressure is 1 atm- **(d) \(273^{\circ}\text{C}, 2 \text{ atm}\)**: Temperature is 546 K, pressure is 2 atm.

Step 4: Comparing the Conditions

From Step 3, we can see that condition (b), with the highest pressure of 2 atm at the same relatively low temperature of 273 K, will result in the highest density according to our \( \rho \propto \frac{P}{T} \) relationship. The increase in pressure raises the density, while the low temperature keeps it from diluting too quickly.

Key Concepts

Gas Density

Gas density is the measure of how compact the gas particles are within a certain space. It is essentially the mass of the gas divided by its volume. Understanding gas density is especially useful in various scientific calculations and processes. When dealing with gases, the density can be effectively calculated using the Ideal Gas Law. The equation for this law is given by:\( PV = nRT \)Here:

• \( P \) is the pressure of the gas,
• \( V \) is the volume occupied by the gas,
• \( n \) is the number of moles,
• \( R \) is the universal gas constant,
• \( T \) is the temperature in Kelvin.

To express density \( \rho \) in terms of these variables, use the formula:\[ \rho = \frac{PM}{RT} \]where \( M \) is the molar mass of the gas. This relationship reveals how gas density is affected by pressure and temperature changes. By increasing pressure or decreasing temperature, the density of the gas increases as gas particles become more compact.

Pressure and Temperature Relationship

The pressure and temperature relationship is a fundamental concept in understanding the behavior of gases. According to the Ideal Gas Law, pressure and temperature greatly influence how gases behave. The formula \( \rho = \frac{PM}{RT} \) demonstrates that:

• Gas density increases with higher pressure, as pressure acts to compress the gas particles more tightly together.
• Conversely, gas density decreases with higher temperature, since heat causes the gas particles to move more energetically and spread out.

This dynamic is an inverse relationship between temperature and density. At higher temperatures, the kinetic energy of gas molecules increases, causing them to move more rapidly, which results in expansion and a decrease in density.Understanding this relationship is crucial when trying to predict how a gas will react under changing environmental conditions. The conditions mentioned, such as elevated pressure with moderate temperature, will result in a relatively high gas density due to the compressive power of pressure countered by minimal thermal expansion.

Standard Temperature and Pressure (STP)

Standard Temperature and Pressure, often abbreviated as STP, is a reference point used in chemistry to define a set of conditions. It enables scientists to measure and compare data on gases in a consistent manner. STP is defined as a temperature of 273 K (which is equivalent to 0°C) and a pressure of 1 atmosphere (atm). At STP, gases follow predictable behavior according to the Ideal Gas Law because these conditions eliminate variables and external influences. This makes STP an ideal baseline in scientific study and experimentation. One practical aspect of using STP is that it sets a known density for gases, aiding in calculations and predictions. At this state, gases are neither under compressive forces nor extensive thermal influence, thus behaving in a stable and predictable manner. This predictability makes STP a crucial concept in both academic and practical chemical applications.