Problem 60 - A Complete Resource Book in Chemistry for JEE Main

Question

Specific impulse of a rocket and the critical temperature of the fuel reacted in the motor of the rocket has the relationship. (a) \(\mathrm{I}_{\mathrm{s}} \propto \sqrt{(\mathrm{T} c)}\) (b) \(\mathrm{I}_{\mathrm{s}} \propto \mathrm{Tc}\) (c) \(\mathrm{I}_{\mathrm{s}} \propto \sqrt{(1 / \mathrm{Tc})}\) (d) \(I_{s} \propto 1 / T_{c}\)

Step-by-Step Solution

Answer

The correct relationship is given by option (a): \(I_s \propto \sqrt{(T_c)}\).

Step 1: Understand Specific Impulse

Specific impulse (\(I_s\)) is a measure of the efficiency of rocket propulsion. It is defined as the thrust produced per unit weight flow rate of the propellant. It indicates how much thrust can be derived from a given amount of propellant.

Step 2: Interpret Problem Statement

The problem is asking us to identify the relationship between specific impulse (\(I_s\)) and critical temperature (\(T_c\)) of the fuel. We need to analyze the given options to find the correct expression that describes this relationship.

Step 3: Analyze Option A

Option (a) states \(I_s \propto \sqrt{(T_c)}\). This suggests that specific impulse increases with the square root of the critical temperature. This matches the known relation that \(I_s\) is proportional to the square root of the temperature at which the reaction occurs because temperature increases lead to faster particle motion and better impulse.

Step 4: Analyze Option B

Option (b) states \(I_s \propto T_c\). This option suggests a direct proportionality between specific impulse and critical temperature, which is not consistent with theoretical relationships.

Step 5: Analyze Option C

Option (c) states \(I_s \propto \sqrt{(1 / T_c)}\). This implies that specific impulse would increase as the critical temperature decreases, which is contrary to physical principles since increasing temperature generally increases specific impulse.

Step 6: Analyze Option D

Option (d) states \(I_s \propto 1 / T_c\). This implies that specific impulse is inversely proportional to the critical temperature. This is incorrect as higher temperatures typically enhance propulsion efficiency, not reduce it.

Step 7: Select the Correct Option

Option (a) correctly describes the relationship \(I_s \propto \sqrt{(T_c)}\) because increasing the combustion temperature generally results in greater specific impulse, as it increases the velocity of exhaust gases.

Key Concepts

Rocket Propulsion Efficiency

Rocket propulsion efficiency is crucial for determining how effectively a rocket can convert fuel into thrust. This revolves around the concept of specific impulse, denoted as \(I_s\). A higher specific impulse means a rocket can achieve greater thrust from the same amount of propellant.

• Thrust Production: This is the force that moves the rocket forward. The efficiency of generating this force from the consumed propellant defines the propulsion efficiency.
• Specific Impulse: It measures the thrust produced per unit of propellant. It is akin to fuel mileage in cars - higher values signify better efficiency.
• Relation to Velocity: Specific impulse relates to the velocity of the exhaust gases; higher velocity means more efficient propulsion, leading to greater specific impulse.

This concept is integral in designing rocket engines that maximize thrust while minimizing fuel consumption, ensuring rockets are more efficient and capable of longer or higher missions.

Critical Temperature

The critical temperature \((T_c)\) of rocket fuel refers to the temperature at which combustion reactions occur optimally, producing the desired energy for propulsion. The relationship between critical temperature and other factors such as specific impulse is crucial.

• Temperature and Reaction Rate: Higher temperatures often increase the rate of chemical reactions, producing more vigorous propulsion.
• Optimal Combustion: Operating around the critical temperature ensures efficient energy release from fuel, supporting maximal thrust.
• Material Constraints: The materials used in a rocket must withstand these high temperatures to maintain structural integrity.

Understanding the critical temperature helps engineers design fuel compositions and cooling systems that safely maximize rocket performance.

Thrust and Propellant Relationship

Thrust and propellant relationship defines how the type and amount of propellant influence the thrust force produced by a rocket.

• Quantity of Propellant: More propellant generally means more potential thrust; however, this is balanced against the efficiency of the rocket engine.
• Propellant Type: Different propellants burn at varying rates and temperatures, affecting the overall thrust a rocket can achieve.
• Efficiency Considerations: Ideally, a rocket should use less propellant for more thrust, translating to higher efficiency and longer mission capabilities.

This relationship is pivotal in space missions, as it governs how far a rocket can travel and what payloads it can carry. Thorough understanding facilitates improved design and operational strategies for future rocket launches.