Answer

Verified

389.4k+ views

**Hint:**Helium is an inert gas so it’s the closest to an ideal gas. The graph depicts the working of an engine which converts the heat into work, in which heat is given to the gas and then work is done by the gas. The performance of the engine is determined through efficiency. Efficiency is the ratio of work done to the heat given.

Formulas used:

$\eta =\dfrac{W}{Q}$

$Q=n{{C}_{v}}\Delta T+n{{C}_{P}}\Delta T$

**Complete answer:**

Efficiency is used to analyse the performance of an engine. Efficiency is the ratio of work done by the gas to heat given to the gas. Therefore,

$\eta =\dfrac{W}{Q}$ - (1)

Here, $\eta $ is the efficiency

$W$ is the total work done by the gas

$Q$ is the total heat given to the gas

From the figure given above,

The heat given to the gas= heat given to gas from A to B at constant volume+ heat given to the gas from B to C at constant pressure

Therefore,

$Q=n{{C}_{v}}\Delta T+n{{C}_{P}}\Delta T$ - (2)

Here, $n$ is the number of moles

${{C}_{v}}$ is the specific heat at constant volume

${{C}_{P}}$ is the specific heat at constant pressure

$\Delta T$ is change in temperature

From the ideal gas equation,

$PV=nRT$

Here, $P$ is the pressure

$V$ is the volume

$R$ is the gas constant

$T$ is the temperature

From the above equation, we have,

$\Delta PV=nR\Delta T$ (at constant volume) - (3)

Using values from graph from A to B in eq (3)

$\begin{align}

& (2{{P}_{0}}-{{P}_{0}}){{V}_{0}}=nR\Delta T \\

& \Rightarrow \Delta T=\dfrac{{{P}_{0}}{{V}_{0}}}{nR} \\

\end{align}$

Using values from graph B to C

$\begin{align}

& (2{{V}_{0}}){{P}_{0}}=nR\Delta T \\

& \Rightarrow \Delta T=\dfrac{2{{P}_{0}}{{V}_{0}}}{nR} \\

\end{align}$

From monatomic gases, ${{C}_{P}}=\dfrac{5R}{2}$ and ${{C}_{V}}=\dfrac{3R}{2}$

Substituting the values in eq (2), we get,

$\begin{align}

& Q=n{{C}_{v}}\Delta T+n{{C}_{P}}\Delta T \\

& \Rightarrow Q=n\dfrac{3R}{2}\dfrac{{{P}_{0}}{{V}_{0}}}{nR}+n\dfrac{5R}{2}\dfrac{2{{P}_{0}}{{V}_{0}}}{nR} \\

& \Rightarrow Q=\dfrac{3}{2}{{P}_{0}}{{V}_{0}}+5{{P}_{0}}{{V}_{0}} \\

& \therefore Q=\dfrac{13}{2}{{P}_{0}}{{V}_{0}} \\

\end{align}$

The work done by the gas = area under the curve

$\begin{align}

& area=l\times b \\

& \Rightarrow area={{P}_{0}}{{V}_{0}} \\

\end{align}$

Substituting work done and heat given in eq (1), we get,

$\begin{align}

& \eta =\dfrac{W}{Q} \\

& \Rightarrow \eta =\dfrac{{{P}_{0}}{{V}_{0}}}{\dfrac{13}{2}{{P}_{0}}{{V}_{0}}} \\

\end{align}$

$\eqalign{

& \Rightarrow \eta = \dfrac{2}{3} \times 100{\text{ % }} \cr

& \therefore \eta = 15.4{\text{ % }} \cr} $

Therefore, the efficiency of the helium gas is $15.4%$.

**Hence, the correct option is (A).**

**Note:**

The carnot engine has the maximum efficiency out of all engines. The reversible engines act as refrigerators and their efficiency is calculated by the coefficient of performance. At constant pressure, the gas expands when heat is given to it and hence volume increases while at constant volume, the pressure of the gas increases.

Recently Updated Pages

What number is 20 of 400 class 8 maths CBSE

Which one of the following numbers is completely divisible class 8 maths CBSE

What number is 78 of 50 A 32 B 35 C 36 D 39 E 41 class 8 maths CBSE

How many integers are there between 10 and 2 and how class 8 maths CBSE

The 3 is what percent of 12 class 8 maths CBSE

Find the circumference of the circle having radius class 8 maths CBSE

Trending doubts

Which are the Top 10 Largest Countries of the World?

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

One cusec is equal to how many liters class 8 maths CBSE

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

One Metric ton is equal to kg A 10000 B 1000 C 100 class 11 physics CBSE

Change the following sentences into negative and interrogative class 10 english CBSE