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Thus, contrary to the initial perception that: "According to my intuition both Pressure P and Temperature T would increase." It is quite different. Think like it is a metal high pressure gas canister for a fixed volume of gas, like for soda production. A release of gas decreases pressure and the volume remains fixed. Also, Temperature does not increase. Rather, because this is Joule work that the gas performs, and each molecule inside the container retains its average energy, the Temperature remains the same.

$$P_{\text{inside the gas container}} \times V_{\text{essentially constant in the container}} = $$ $$=\left(n_1 - \delta n \right)_{\text{inside the container}} \times R_{1\text{ d}} \times T_{\text{essentially constant in the container due to Joule expansion from the container}}$$

Thus, contrary to the initial perception that: "According to my intuition both Pressure P and Temperature T would increase." It is quite different. Think like it is a metal high pressure gas canister for a fixed volume of gas, like for soda production. A release of gas decreases pressure in the tank and the volume remains fixed. Of course the room is large and its temperature is not changed significantly with a small leak of gas from the canister. Rather the pressure inside the room increases if the room is sealed (aside from the gas leak of the gas canister) in a proportional manner to how many moles of gas are added to the room from the gas canister.

Also, Temperature does not increase. Rather, because this is Joule work that the gas performs, and each molecule inside the container retains its average energy, the Temperature remains the same.

$$P_{\text{inside the gas container}} \times V_{\text{essentially constant in the container}} = $$ $$=\left(n_1 - \delta n \right)_{\text{inside the container}} \times R_{1\text{ d}} \times T_{\text{essentially constant in the container due to Joule expansion from the container}}$$

I am only calculating the parameters of the gas inside the tank because the relatively low pressure of the room are not deemed by the question to be significant enough to include there within. And for most gas tanks at high pressure, the atmospheric pressure and temperature only have a nominal influence, like in the case of gas (only gas, no liquid) for cooking food, for instance.

Thus, taking account that $n_2=n_1-\delta n$ and $P$ is essentially constant and also $T$ is essentially constant, we have:

Thus, taking account that $n_2=n_1-\delta n$ and $V$ is essentially constant and also $T$ is essentially constant, we have:

From this equation one can see that the average internal energy of each molecule does not change, and rather it is the loss of molecules $\delta n$ during the gas release from the container that changes the complete energy stored in the container. The temperature $T$ is related to the average energy of each ideal gas molecule in the gas container, not to how many gas molecules there are. Thus, $T$ is constant, and total energy loss in the container is just the average energy per molecule times the number of molecules.

From this equation one can see that the average internal energy of each molecule does not change, and rather it is the loss of molecules $\delta n$ during the gas release from the container that changes the complete energy stored in the container. The temperature $T$ is related to the average energy of each ideal gas molecule in the gas container, not to how many gas molecules there are. Thus, $T$ is constant, and total energy loss in the container is just the average energy per molecule times the number of molecules.