Problem 121 - A Complete Resource Book in Chemistry for JEE Main

Question

According to Charle's Law (a) \(\left(\frac{\mathrm{dV}}{\mathrm{dT}}\right)_{\mathrm{P}}=\mathrm{k}\) (b) \(\left(\frac{\mathrm{dT}}{\mathrm{d} \mathrm{V}}\right)_{\mathrm{P}}=\mathrm{k}\) (c) \(\left[\frac{1}{T}-\frac{V}{T^{2}}\right]_{p}=0\) (d) \(\mathrm{V} \alpha \frac{1}{\mathrm{~T}}\)

Step-by-Step Solution

Answer

Option (a) is correct according to Charles's Law.

Step 1: Understand Charles's Law

Charles's Law states that at constant pressure, the volume of a gas is directly proportional to its temperature when measured in Kelvin. This is mathematically expressed as \( V \propto T \).

Step 2: Identify the Mathematical Expression

The proportionality \( V \propto T \) can be expressed as \( V = kT \), where \( k \) is a constant of proportionality under constant pressure.

Step 3: Interpret the Statements

Evaluate each given statement according to Charles's Law: - (a) \( \left(\frac{\mathrm{dV}}{\mathrm{dT}}\right)_{\mathrm{P}}=\mathrm{k} \) implies that the rate of change of volume with temperature at constant pressure is constant. This statement aligns with Charles's Law, since \( V = kT \) gives \( \frac{\mathrm{dV}}{\mathrm{dT}} = k \).- (b) \( \left(\frac{\mathrm{dT}}{\mathrm{dV}}\right)_{\mathrm{P}}=\mathrm{k} \) implies an inverse relationship between temperature and volume, which contradicts Charles's Law.- (c) \( \left[\frac{1}{T}-\frac{V}{T^{2}}\right]_{p}=0 \) simplifies using \( V = kT \) to \( 0 = 0 \), validating the equation under Charles's Law.- (d) \( \mathrm{V} \alpha \frac{1}{\mathrm{~T}} \) suggests that volume is inversely proportional to temperature, which is contrary to Charles's Law.

Step 4: Determine Correct Option

By analyzing each statement:- (a) correctly aligns with Charles's Law as \( V = kT \).- (b) indicates incorrect inverse relationship.- (c) mathematically supports \( V = kT \) by reducing to a true statement when substituted.- (d) contradicts the initial premise by suggesting inverse proportionality.Thus, statement (a) is correct.

Key Concepts

Gas Laws

Understanding gas behavior requires insight into various gas laws. These laws describe how gases respond to different conditions like pressure, volume, and temperature.
Charles's Law is one of these fundamental gas laws and specifically focuses on the volume-temperature relationship.

Gas laws help explain behaviors in everyday life. For example, Charles's Law explains why balloons expand when heated. Keep in mind that these laws assume ideal conditions where gases behave perfectly without interaction between particles.

• Boyle’s Law: Relates pressure and volume at constant temperature ( P V = k )
• Charles’s Law: Relates volume and temperature at constant pressure ( V ∝ T )
• Gay-Lussac’s Law: Relates pressure and temperature at constant volume ( P ∝ T )

Grasping these laws is essential for solving chemistry problems related to gases. They form the basis for more complex thermodynamic equations and principles.

Proportionality in Thermodynamics

Proportionality is a key concept in thermodynamics, which is the study of energy and heat transfer.
In the case of Charles's Law, volume and temperature are directly proportional. This means as temperature ( T ) increases, volume ( V ) also increases, assuming constant pressure.

Understanding proportionality helps you predict how changes in one variable affect another. In Charles's Law, this can be represented by the equation V = kT , where k is the constant.
Conversely, if either temperature or volume decreases, the other will also decrease to maintain the relationship.

• Direct Proportionality: Both variables move in the same direction.
• Inverse Proportionality: One variable increases while the other decreases, not applicable in Charles’s Law but in other contexts like when considering statement (d) from the exercise.

Proportionality simplifies complex concepts into more understandable small parts. Use this understanding to approach other problems involving gas laws.

Volume and Temperature Relationship

The relationship between volume and temperature is core to Charles's Law. When we say volume and temperature are directly proportional, we mean that if you increase the temperature of a gas, its volume will increase, provided the pressure stays the same.
The equation V = kT is the mathematical representation of this idea, where k is a constant. This ensures that the proportion between volume and temperature remains steady.

Consider a balloon. When heated, the air inside gains energy; particles move more quickly and push outward, expanding the balloon. This response showcases the direct relationship of volume and temperature.
The relationship breaks down if the pressure is not constant, leading to different outcomes. Always control variables in experiments and exercises to align with theoretical predictions.
Applying this law effectively requires converting temperatures into Kelvin. Since Kelvin is proportional to absolute temperature scales, it prevents negative numbers, simplifying calculations.

Chemistry Problem Solving

Effective chemistry problem solving involves understanding and applying scientific concepts correctly.
To solve problems related to Charles's Law, begin by identifying all given information and what is being asked.

Recognize that Charles's Law requires constant pressure. Ensure the temperature is in Kelvin to maintain direct proportionality with volume.
Carefully set up equations based on the relationship V = kT .
Let’s look at a strategy:

• Read and understand the problem thoroughly.
• List known variables and constants.
• Convert all temperatures to Kelvin.
• Use the equation V = kT to find unknown variables.
• Cross-check calculations for errors.

Understanding the core principles behind problems, like the relationship between volume and temperature, can enhance problem-solving skills and lead to more accurate answers. Practice consistently to improve your ability to apply these concepts in diverse scenarios.