Question

One mole of an ideal monatomic gas at temperature \[{{T}_{\text{o}}}\]expands slowly according to the law \[\dfrac{P}{V}=\text{constant}\]. If the final temperature is 2Tₒ, heat supplied to the gas is?
$
\text{A}\text{. }2R{{T}_{0}} \\
\text{B}\text{. }R{{T}_{0}} \\
\text{C}\text{. }\dfrac{3}{2}R{{T}_{0}} \\
\text{D}\text{. }\dfrac{1}{2}R{{T}_{0}} \\
$

Answer 1

Hint: The gas expands according to the law \[\dfrac{P}{V}=\text{constant}\], so the gas undergoes polytropic process and its heat capacity remains constant.

Formula used:
The equation of a polytrope is given by
 \[P{{V}^{x}}=\text{constant}\]
  Then, heat capacity of the gas for polytropic process is
\[c={{c}_{v}}+\dfrac{R}{1-x}\]
where, cᵥ = heat capacity at constant volume, R = gas constant.
And, amount of heat supplied to the gas is given by,
\[Q\text{ }=\text{ }nc\Delta T\] where, n (i.e., the amount of substance) = 1 [since, monoatomic].

Complete step by step answer:
Given: \[\dfrac{P}{V}=\text{constant}\]
    => \[P{{V}^{-1}}\text{ }=\text{ constant}\]
 Now, comparing the above equation with the equation of polytrope , i.e. \[P{{V}^{x}}=\text{constant}\],
 \[x=-1\]
Now, consider the heat capacity equation \[c={{c}_{v}}+\dfrac{R}{1-x}\]
Substituting the value \[x=-1\] , and \[{{c}_{v}}=\dfrac{3}{2}R\] (monoatomic) in the heat capacity formula:
  \[c={{c}_{v}}+\dfrac{R}{1-x}=\dfrac{3}{2}\text{R+}\dfrac{R}{2}\text{= }2R\]
 The change in temperature is:
\[\Delta T=2{{T}_{\text{o}}}-{{T}_{\text{o}}}={{T}_{\text{o}}}\]
Now, substituting the value \[c=2R\], \[n=1\] (monoatomic) , and \[\Delta T={{T}_{\text{o}}}\] in the heat-equation:
  \[Q\text{ }=\text{ }nc\Delta T=\text{ }1~\text{x}(2R)\text{x}(2{{T}_{0}}-{{T}_{0}}~)=2R{{T}_{0}}\]

So, the correct answer is “Option A”.

Additional Information:
The ideal gas equation is also called the general gas equation which is a theoretical gas composed of randomly moving point particles which are generally not subjected to intermolecular interactions. The ideal gas concept obeys the ideal gas law which is a simplified equation of state and is susceptible to analysis under statistical mechanics.

Note:
It is tough to exactly describe the real gas, so the concept of ideal gas with certain assumptions were created which is an approximation that helps to predict the behavior of real gas. The assumptions are:
1) Gases are made up of molecules which are in random motion in straight lines.
2) The molecules behave like rigid spheres.
3) Pressure created due to the collision between the walls of the container and the molecules.
4) All the collisions between the walls of the container and the molecules are perfectly elastic.
5) The gas temperature is proportional to the average kinetic energy of the molecules.