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Hint: To solve this question, look at the parameters given to us in the question. Since, pressure (P), volume (V) and real gas constant (R) remain constant, relate the number of moles and temperature by using the ideal gas equation.

Complete step by step answer:

According to the question, there is an open vessel with a temperature equal to 27 deg Celsius.

27\[{}^\circ C\] = 273.15 + 27 = 300K.

Also, two-fifth of air escapes. Therefore, we can say that there is a change in moles of the gas.

So, let the initial moles in air be ‘n1’ and the number of moles after two fifths of gas escaped be ‘n2’.

Let n1 = 1 mole

So, n2 = n1 – (2/5) = 1 – (2/5) = 3/5 moles.

According to the question we can say that the volume remains constant.

Ideal gas equation relates PV = nRT.

Since, pressure (P), volume (V) and real gas constant (R) are constant, we can relate the number of moles and temperature as –

nT = constant

\[{{n}_{1}}{{T}_{1}}={{n}_{2}}{{T}_{2}}\]= constant

Now, putting the values of moles and temperature we get –

\[\begin{align}

& (1)(300)=\left( \dfrac{3}{5} \right){{T}_{2}} \\

& {{T}_{2}}=\dfrac{\text{300x}5}{3}K \\

& {{T}_{2}}=500K \\

\end{align}\]

Therefore, the answer is – option (d). The temperature at which the vessel has been heated is 500K.

Additional Information:

1 mole of any gas at STP occupies a volume of 22.4 L.

Note: Ideal gas equation is the equation of state of an ideal gas (hypothetical). It is an approximation of the behaviour of gases under ideal conditions.It is a combination of empirical laws like Boyle’s law, Charles law, Gay-Lussac’s law and Avogadro’s law.

Complete step by step answer:

According to the question, there is an open vessel with a temperature equal to 27 deg Celsius.

27\[{}^\circ C\] = 273.15 + 27 = 300K.

Also, two-fifth of air escapes. Therefore, we can say that there is a change in moles of the gas.

So, let the initial moles in air be ‘n1’ and the number of moles after two fifths of gas escaped be ‘n2’.

Let n1 = 1 mole

So, n2 = n1 – (2/5) = 1 – (2/5) = 3/5 moles.

According to the question we can say that the volume remains constant.

Ideal gas equation relates PV = nRT.

Since, pressure (P), volume (V) and real gas constant (R) are constant, we can relate the number of moles and temperature as –

nT = constant

\[{{n}_{1}}{{T}_{1}}={{n}_{2}}{{T}_{2}}\]= constant

Now, putting the values of moles and temperature we get –

\[\begin{align}

& (1)(300)=\left( \dfrac{3}{5} \right){{T}_{2}} \\

& {{T}_{2}}=\dfrac{\text{300x}5}{3}K \\

& {{T}_{2}}=500K \\

\end{align}\]

Therefore, the answer is – option (d). The temperature at which the vessel has been heated is 500K.

Additional Information:

1 mole of any gas at STP occupies a volume of 22.4 L.

Note: Ideal gas equation is the equation of state of an ideal gas (hypothetical). It is an approximation of the behaviour of gases under ideal conditions.It is a combination of empirical laws like Boyle’s law, Charles law, Gay-Lussac’s law and Avogadro’s law.

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