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Question: An ideal gas expands in such a way that \(P{V^2}\) = constant throughout the process. A. The graph...

An ideal gas expands in such a way that PV2P{V^2} = constant throughout the process.
A. The graph of the process of T-V diagram is a parabola
B. The graph of the process of T-V diagram is a straight line
C. Such an expansion is possible only with heating
D. Such an expansion is possible only with cooling

Explanation

Solution

Use the ideal gas equation PV=nRTPV = nRT to find the relation.Ideal gas equation can be obtained by combining Boyle's law and Charle’s law. As PV2P{V^2}=constant, multiply both sides of the equation by V to find the result.
Complete answer:
Boyle’s law states that if temperature remains constant, the volume V of the given mass of a gas is inversely proportional to the pressure, i.e. V∝1PV \propto \dfrac{1}{P} or PV = K, where K is a constant whose value depends upon the nature and temperature of the gas.

Charle’s law states that if pressure remains constant, the volume of a given mass of a gas increases or decreases by 1273.15\dfrac{1}{{273.15}} of its volume at 0° for each 1°C rise or fall in temperature, i.e. V∝TV \propto T.

From the ideal gas equation, we havePV=nRTPV = nRT, Where, P is the pressure, V is the volume, T is the temperature and R is the universal gas constant.
It is given that PV2P{V^2} = constant throughout the process.
So, multiplying PV=nRTPV = nRT by V on both sides, we get
PV2=nVRT⇒nVRT=constantP{V^2} = nVRT \Rightarrow nVRT = constant

For the given gas n and R are constant. Therefore, we can conclude that
VT=constantVT = constant
It implies that V is inversely proportional to the temperature. So, when volume increases, temperature decreases. This is possible only when the gas is cooled.

Hence, the correct options are B and D.

Note: Look out for the given relation and the constants. Try to eliminate the given options on the given facts. As VT = constant, so options C and D can be easily eliminated.In thermodynamics, Ideal gas law is a well-defined approximation of the behavior of many gases under diverse conditions. Ideal Gas Equation is the combination of empirical laws like Charle’s law, Boyle’s law, Gay-Lussac’s law, and Avogadro’s law.