Question

An ideal gas in a thermally insulated vessel at internal pressure = ${P_1}$, volume = ${V_1}$ and absolute temperature = ${T_1}$ expands irreversibly against zero external pressure, as shown in the diagram. The final internal pressure, volume and absolute temperature of the gas are ${P_2}$, ${V_2}$ and ${T_2}$ respectively. For this expansion,
a.) q = 0
b.) ${T_2}$ = ${T_1}$
c.) ${P_2}$ ${V_2}$ = ${P_1}$ ${V_1}$
d.) ${P_2}$ $V_2^\Upsilon$ = ${P_1}$ $V_1^\Upsilon$

Answer 1

The gas is present in a thermally insulated vessel. There is no change in heat and even the process is further isothermal. These points explain all the correct options present. There no change in heat does not signify the process being adiabatic.

Complete step by step answer:
In the question, we are given that the ideal gas is present in a thermally insulated vessel.
We know that in a thermally insulated vessel, there is no change in mass and temperature. So, this means that the number of moles will remain constant.
Further, there is no change in heat. Thus, q = 0.
So, the option a.) is correct.
Another thing, we get to know is that thermally insulated. This means the change in temperature is also not there. This means, ${T_2}$ = ${T_1}$.
So, the option b.) is also correct.
Further, we know that PV = nRT .
In the given conditions, the number of moles i.e. ‘n’ is const, temperature is const and R is already gas const. So, the right hand side is const.
There is change in only pressure and volume in the above reaction.
So, ${P_2}$ ${V_2}$ = ${P_1}$ ${V_1}$
Thus, the option c.) is also correct.
Now, if we see the option d.) which is ${P_2}$ $V_2^\Upsilon$ = ${P_1}$ $V_1^\Upsilon$. This equation is for adiabatic processes. There is nothing mentioned about the adiabatic process in this question. Thus, the option d.) is incorrect.
So, the correct answer is “Option A, B and C”.

Note: It must be noted that the word insulated means there is no flow of things from one side to another. The thermal insulation means there is even no flow of heat. This means the process is isothermal. We know that in the isothermal process there is only change in pressure and volume of the substances.