Question

An ideal gas undergoes a process in which \[T = {T_0} + a{V^3}\] , where \[{T_0}\] and ‘a’ are positive constants and V is Molar volume. The molar volume for which the pressure will be minimum is:
(A) \[{(\dfrac{{{T_0}}}{{2a}})^{\dfrac{1}{3}}}\]
(B) \[{(\dfrac{{{T_0}}}{{3a}})^{\dfrac{1}{3}}}\]
(C) \[{(\dfrac{a}{{2{T_0}}})^{\dfrac{1}{3}}}\]
(D) None of the above

Answer 1

Hint: An ideal is a theoretical concept which states the existence of a perfectly ideal gas in which the gas molecules are not subject to interparticle interaction, even when the nature of their movements is random. On the other hand, real gases are those gases that do not exhibit this ideal behaviour and have deviations in the values of their characteristic properties.

Complete Step-by-Step Answer:
Before we forward with the solution of this question, let us understand a few basic important concepts.
These deviations in real gases can be caused by various physical as well as chemical characteristics of the gas. The properties of ideal gases can be determined by an equation which establishes the ideal relations between the properties like pressure, volume, temperature and the number of moles of the gas present. The mathematical representation of this equation is given as:
PV = nRT, where
P = pressure
V = volume
 n = number of moles
T = temperature
 R = gas constant (constant of proportionality)
In the question, the value of T has been given as: \[T = {T_0} + a{V^3}\]
Substituting this value in the Ideal Gas Equation, we get:
PV = nR ( \[{T_0} + a{V^3}\] )
 \[{T_0} + a{V^3} = \dfrac{{PV}}{{nR}}\]
P = \[\dfrac{{({T_0} + a{V^3})nR}}{V}\]
P = \[\dfrac{{({T_0}nR) + (nRa{V^3})}}{V}\]
Now, another condition that has been to us is that the pressure should be minimum. To satisfy this condition, the value of \[\dfrac{{dP}}{{dV}}\] must be equal to zero.
Hence, differentiating both sides of the above equation with respect to V, we get
 \[\dfrac{{dP}}{{dV}}\] = \[\dfrac{{ - nR{T_0}}}{{a{V^2}}} + 2nRaV\]
0 = \[\dfrac{{ - nR{T_0}}}{{a{V^2}}} + 2nRaV\]
V = \[{(\dfrac{{{T_0}}}{{2a}})^{\dfrac{1}{3}}}\]

Hence, Option A is the correct option

Note: Real fluids at low density and high temperature approximate the behaviour of a classical ideal gas. However, at lower temperatures or a higher density, a real fluid deviates strongly from the behaviour of an ideal gas, particularly as it condenses from a gas into a liquid or as it deposits from a gas into a solid.