Question

Specific impulse of a rocket

• A. is proportional to combustion chamber temperature
• B. is inversely proportional to square root of molecular weight of combustion products
• C. is proportional to molecular weight of combustion products
• D. is proportional to square root of molecular weight of combustion products

Hint:

For rocket design, choosing propellant with a low molecular weight can significantly enhance the specific impulse, improving the overall efficiency of the rocket.

Answer 1

The Correct Option is B

Step 1: Understand Specific Impulse.
Specific impulse (\(I_{sp}\)) of a rocket engine is a measure of how efficiently a rocket uses the mass of its propellant. It is defined as the thrust per unit mass flow rate of propellant, typically given in seconds. Mathematically, it is expressed as: \[ I_{sp} = \frac{v_e}{g_0} \] where \(v_e\) is the effective exhaust velocity, and \(g_0\) is the standard gravity.
Step 2: Relate to Exhaust Velocity and Molecular Weight.
The exhaust velocity (\(v_e\)), from the rocket propulsion theory, is influenced by the combustion chamber temperature (\(T_c\)) and the molecular weight (\(M\)) of the exhaust gases: \[ v_e = \sqrt{\frac{2kRT_c}{M}} \] where \(k\) is the specific heat ratio, \(R\) is the universal gas constant, and \(T_c\) is the combustion temperature.
Step 3: Derive Specific Impulse Relation.
Given the relationship for \(v_e\), the specific impulse can be rewritten as: \[ I_{sp} = \frac{\sqrt{\frac{2kRT_c}{M}}}{g_0} \] It is evident from this formula that \(I_{sp}\) is directly proportional to the square root of the combustion chamber temperature and inversely proportional to the square root of the molecular weight of the combustion products. This highlights that the specific impulse increases with higher temperatures and decreases with higher molecular weights of the exhaust gases.