Considering the ideal gas law, the ratio of the final to the initial number of moles in the container is 4.
An ideal gas is a theoretical gas that is considered to be composed of point particles that move randomly and do not interact with each other. Gases in general are ideal when they are at high temperatures and low pressures.
An ideal gas is characterized by three state variables: absolute pressure (P), volume (V), and absolute temperature (T). The relationship between them constitutes the ideal gas law, an equation that relates the three variables if the amount of substance, number of moles n, remains constant and where R is the molar constant of the gases:
P× V = n× R× T
Then ,the number of moles can be expressed as:
[tex]n=\frac{PxV}{RxT}[/tex]
In this case, a container filled with an ideal gas is connected to a reservoir of the same gas so that the number of moles in the container can change. You want to know the ratio of the final to the initial number of moles in the container, this is, [tex]\frac{n_{final} }{n_{initial} }[/tex].
The number of moles [tex]n_{initial}[/tex] can be expressed as:
[tex]n_{initial} =\frac{P_{initial} xV_{initial} }{RxT_{inital} }[/tex]
and the number of moles [tex]n_{final}[/tex] can be expressed as:
[tex]n_{final} =\frac{P_{final} xV_{final} }{RxT_{final} }[/tex]
Then:
[tex]\frac{n_{final} }{n_{initial} } =\frac{\frac{P_{final} xV_{final} }{RxT_{final} }}{\frac{P_{initial} xV_{initial} }{RxT_{initial} }}[/tex]
Being R is a constant number:
[tex]\frac{n_{final} }{n_{initial} } =\frac{P_{final} xV_{final} }{T_{final} }x\frac{T_{initial} }{P_{initial} xV_{initial} }[/tex]
The pressure and volume of the container are each doubled while the temperature is held constant, then you know:
Replacing in the the ratio of the final to the initial number of moles:
[tex]\frac{n_{final} }{n_{initial} } =\frac{2xP_{initial} x2xV_{initial} }{T_{initial} }x\frac{T_{initial} }{P_{initial} xV_{initial} }[/tex]
[tex]\frac{n_{final} }{n_{initial} } =\frac{4xP_{initial} xV_{initial} }{T_{initial} }x\frac{T_{initial} }{P_{initial} xV_{initial} }[/tex]
[tex]\frac{n_{final} }{n_{initial} } =4}[/tex]
Finally, the ratio of the final to the initial number of moles in the container is 4.
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