Deriving the Temperature-Volume Relation for \(PV^2 = \text{constant}\)
We are given a process for an ideal gas where the relation between pressure (\(P\)) and volume (\(V\)) is \(PV^2 = K\), where \(K\) is a constant.
From the Ideal Gas Law, we know that for a fixed amount of gas (\(n\)), \(PV = nRT\), where \(R\) is the ideal gas constant and \(T\) is the absolute temperature.
We can express pressure (\(P\)) from the Ideal Gas Law as: \(P = \frac{nRT}{V}\)
Now, substitute this expression for \(P\) into the given process equation (\(PV^2 = K\)): \( \left(\frac{nRT}{V}\right) V^2 = K \) \( nRT V = K \)
Since \(n\) and \(R\) are constants, we can rewrite the equation as: \( TV = \frac{K}{nR} \) Let \(\frac{K}{nR} = \text{constant}'\). Therefore, the process follows the relation: \( TV = \text{constant}' \)
Applying the Constant Relation to Initial and Final States
This means that the product of temperature and volume remains constant throughout the process. Applying this to the initial state (1) and the final state (2): \( T_1 V_1 = \text{constant}' \) \( T_2 V_2 = \text{constant}' \)
Equating these two expressions: \( T_1 V_1 = T_2 V_2 \)
To find the relationship between the temperature ratio and the volume ratio, rearrange the equation: \( \frac{T_1}{T_2} = \frac{V_2}{V_1} \)
This result matches option C.
Similar Questions
A system that does NOT allow exchange of heat with its surrounding is called
Important Questions from Thermodynamics
A system that does NOT allow exchange of heat with its surrounding is called
- Which of the following statements correctly describes the thermodynamic classification of entropy?
- A mass of $10 \text{ kg}$ is suspended vertically by a rope from the roof. A horizontal force is applied on the rope at a point $P$. The point $P$ is $1 \text{ m}$ vertically below the roof attachment point, and the length of the rope segment from the roof to $P$ is $2 \text{ m}$. If the suspended mass is in equilibrium, what is the tension in the upper part of the rope (from roof to $P$)? (Take $g = 10 \text{ ms}^{-2}$)
Example of thermoplastic among the following is
The fuel with highest calorific value in kJ /kg unit is-
