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Work done of an expanding gas in vacuum

Before, we start discussing about the work done of an expanding gas, it is advisable to first of all know about the work done of a gas.
The work done of a gas is derived as follows:
Recall: the work done (w) is given as:
$$w = fs$$ ........(1)
Where,
f = force
s = distance

But for a gas; work done (w) is given as:
$$w = pv$$ .........(2)
Where, p = pressure and v = volume

During a change of state the work done is written as:
$$dw=pdv$$
Adding all the change, we have:
$$\int dw = \int_{v_1}^{v_2} pdv$$ .......(3)
But, p = constant
Therefore, equ (3) becomes:
$$w = p \int_{v_1}^{v_2} dv$$ .............(4)
Equ(4) is the work done of the gas.

Expansion of gas
Expansion of gas which is sometimes called "free expansion" is a process in which gas expands into an insulated evacuated chamber.
For real gases,  they experience change in temperature but for ideal gases, there is no change in temperature.
From the equation:
$$p_1v_1 = p_2v_2$$
Where;
p1 = initial pressure
p2 = final pressure
v1 = initial volume
v2 = final volume

When gas expands; $$v_2 > v_1$$ and $$p_2 < p_1$$

Work done of an expanding gas in vacuum:
Recall: From expansion of gas that
$$p_2 < p_1$$
hence,
$$p = 0$$ ...............(7)
From equ(4) substitute equ(7)
$$w = 0 \int_{v_1}^{v_2} dv$$
Hence,
$$w=0$$
Therefore,  the work done for an expanding gas in a vacuum is zero

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