Question

Five mole of a diatomic gas is kept at temperature$T$. The volume of the gas varies according to the law, $V = a{T^{ - 2}}$, where a is positive constant. The final temperature of the gas is found to be $5T$, what amount of heat is supplied to the gas?
A. $5RT$
B. $(5/2)RT$
C. $(10/3)RT$
D. $10RT$

Answer 1

As we know that for the diatomic gas, number of molecules are 5, initial temperature is T. Also V varies as $V = a{T^{ - 2}}$ where $a$ is positive constant and also final temperature is 5T. The atomicity is the number of atoms combined to form the molecule to remain in the stable state. Mono-atomic gas are the atoms which have 1 atomicity and diatomic has 2 atomicity.

Complete step by step answer:
Number of mole$= 5mole$
Temperature = $T$ (Initial temperature)
$V = a{T^{ - 2}}$(Here V is the volume of gas)
Final temperature = 5T
As we know that
$\dfrac{{PV}}{T} = {\text{constant = c}}$
So, $\dfrac{{PV}}{T} = {\text{constant = c}}$
$\dfrac{{PV}}{C} = T$
Substitute the value of temperature, we get-
$V = a{T^{ - 2}}$
We can write-
$V{T^2} = a$
$\Rightarrow P \times {\left( {\dfrac{{PV}}{C}} \right)^2} = a$
$\Rightarrow{P^2}{V^3} = {\text{constant}}$
$\Rightarrow P{V^{\dfrac{3}{2}}} = {C_2}$----- (1)

Now compare this equation with $P{V^N} = {\text{Constant}}$, we get-
$N = \dfrac{3}{2}$
Now for finding the value of C (Molar heat capacity)
$C = ({C_V} - \dfrac{R}{{N - 1}})$
$\Rightarrow C = \dfrac{{5R}}{2} - \dfrac{R}{{\dfrac{3}{2} - 1}}$
$\Rightarrow C = \dfrac{R}{2}$
Now, heat supplied,$\Delta Q = n \times C \times \Delta T$
Here $Q$ is the heat supplied, $N$ is the number of moles, $C$ is the heat capacity and ∆t is the change in initial and final temperature.

By substituting all the values, we get,
$5 \times \dfrac{R}{2} \times \left( {5T - T} \right)$
$\Rightarrow 5 \times \dfrac{R}{2} \times \left( {4T} \right)$
$\therefore 10RT$

So, option (D) is correct.

Note: Heat capacity is an extensive property. It is a physical property which is defined as the amount of heat that is supplied to the given mass of material to produce a unit change in its temperature. SI unit of heat capacity is $Joules/k$.It is also known as thermal capacity.