Question

An ideal gas expands in such a way that $P{V^2}$ = 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

Answer 1

Hint: Use the ideal gas equation \[PV = nRT\] to find the relation.Ideal gas equation can be obtained by combining Boyle's law and Charle’s law. As \[P{V^2}\]=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. $V \propto \dfrac{1}{P}$ 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 $\dfrac{1}{{273.15}}$ of its volume at 0° for each 1°C rise or fall in temperature, i.e. $V \propto T$.

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

For the given gas n and R are constant. Therefore, we can conclude that
$VT = constant$
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.